Computing I/Q Impairments at System Output Based on I/Q Impairments at System Input

ABSTRACT

Systems and methods for measuring transmitter and/or receiver I/Q impairments are disclosed, including iterative methods for measuring transmitter I/Q impairments using shared local oscillators, iterative methods for measuring transmitter I/Q impairments using intentionally-offset local oscillators, and methods for measuring receiver I/Q impairments. Also disclosed are methods for computing I/Q impairments from a sampled complex signal, methods for computing DC properties of a signal path between the transmitter and receiver, and methods for transforming I/Q impairments through a linear system.

RELATED APPLICATION DATA

This application is a continuation of U.S. patent application Ser. No.13/442,573, filed on Apr. 9, 2012, entitled “Transforming I/QImpairments Through a Linear System”, invented by Stephen L. Dark, whichis a continuation of U.S. patent application Ser. No. 13/404,896, filedon Feb. 24, 2012, entitled “Techniques for the Measurement of I/QImpairments and Associated Computational Tools”, invented by StephenDark and Christopher Behnke. The above-identified applications arehereby incorporated by reference in their entireties as though fully andcompletely as forth herein.

FIELD OF THE INVENTION

The present invention relates to the field of signal processing, andmore particularly to systems and methods for measurement and correctionof I/Q impairments in a receiver device or a transmitter device.

DESCRIPTION OF THE RELATED ART

A transmitter receives a complex digital signal I(n)+jQ(n), converts thecomplex digital signal into an analog signal I(t)+jQ(t), and upconvertsthe analog signal using an I/Q modulator. The upconverted signal istransmitted onto a channel. Ideally, a pure complex exponential tonesupplied to the I/Q modulator will result in a pure tone beingtransmitted. In reality, however, I/Q impairments in the transmitterwill cause the I channel and Q channel to have different gains anddifferent phase displacements. Such distortions imply among other thingsthat the transmitted signal will have unwanted energy at a frequencyequal to the negative of the tone frequency. Depending on thecommunication standard, this unwanted “image” results in potential skewon the constellation diagram or an artificial noise floor. Receivershave a similar problem. When a receiver is stimulated with a pure toneat frequency f, the complex signal appearing at the output of thereceiver's I/Q demodulator will include unwanted signal energy atfrequency −f in addition to energy at frequency f. In both case(transmitter and receiver), the difficulty arises due to imbalances ingain and phase between the I channel and the Q channel. Thus, thereexists a need for mechanisms capable of correcting for I/Q impairmentsin a transmitter and/or receiver.

Furthermore, to achieve a high-quality correction for I/Q impairments,one needs to have access to a high-quality measurement of the I/Qimpairments. However, quality measurements may be difficult to obtain.For example, to measure the I/Q impairments of a transmitter involvesdirecting the transmitter to transmit a signal to a receiver. Thereceiver estimates the transmitter's I/Q impairments based on itsreceived signal. However, the receiver's I/Q demodulator corrupts theestimates with its own I/Q impairments. Furthermore, the signal pathbetween the transmitter's I/Q modulator and the receiver's I/Qdemodulator also introduces distortion to the estimates. Thus, thereexists a need for mechanisms capable of estimating or measuring the I/Qimpairments of transmitters and/or receivers, for mechanisms capable ofaccurately measuring the I/Q impairments implicit in sampled signals,for mechanisms capable of determining the properties of the signal path,and for mechanisms capable of predicting how I/Q impairments aretransformed by systems such as the signal path.

SUMMARY

Among other things, the present patent discloses mechanisms capable ofcompensating for I/Q impairments in a transmitter and/or a receiver. Theparameters used to perform the compensation are computed based onmeasured or estimated values of the I/Q impairments. For example, theparameters used to compensate for the I/Q impairments of a transmitter(or receiver) are computed based on measured or estimated values ofthose impairments. Any known technique may be used to measure orestimate the I/Q impairments of the transmitter or the receiver or aseries combination of the transmitter and receiver, including, but notlimited to, the techniques disclosed herein.

In one embodiment, a system and method for compensating for I/Qimpairments of a receiver may involve the following operations.

An analog input signal is received from a transmission medium. I/Qdemodulation is performed on the analog input signal to produce ananalog inphase (I) signal and an analog quadrature (Q) signal. Theanalog I signal and the analog Q signal are then digitized to producerespectively a digital I signal and a digital Q signal. The digital Isignal and the digital Q signal are filtered in accordance with a 2×2matrix of digital filters to produce a filtered digital I signal and afiltered digital Q signal. (The filtering may be performed on aprogrammable hardware element such as an FPGA, or in dedicated digitalcircuitry such as an ASIC, or in software on a processor, etc.) The 2×2matrix of digital filters at least partially compensates for I/Qimpairments of the receiver over a range of frequencies. The frequencyresponse of at least one of the diagonal components of the 2×2 matrix iscomputed based on measurements of the I/Q impairments as a function offrequency and the measurements as a function of the negative offrequency. (The measurements of the receiver's I/Q impairments may beobtained by any known method. The present document describes a number ofmethods for obtaining such measurements.) Furthermore, the frequencyresponse of at least one of the non-diagonal components of the 2×2matrix is computed based on the measurements as a function of frequencyand the measurements as a function of the negative of frequency.

In some embodiments, the receiver's I/Q impairments over positivefrequencies and the receiver's I/Q impairments over negative frequenciesmay be assumed to be functionally related. (A) In one such embodiment,the frequency responses of the 2×2 matrix may be computed as follows.The frequency response of at least one of the diagonal components of the2×2 matrix at the arbitrary frequency f may be computed based only on ameasurement of the I/Q impairments at the frequency f (or alternatively,only on a measurement of the I/Q impairments at the frequency −f).Furthermore, the frequency response of at least one of the non-diagonalcomponents of the 2×2 matrix at the frequency f may be computed basedonly on the measurement of the I/Q impairments at the frequency f (oralternatively, only on the measurement of the I/Q impairments at thefrequency −f). (B) In another such embodiment, the gain imbalance isassumed to be even and the phase skew is assumed to be odd. Then bothnon-diagonal components of the 2×2 matrix may be set to zero; one of thediagonal components may correspond to a pure pass-through filter (i.e.,unity frequency response); and a frequency response of the otherdiagonal component at the arbitrary frequency f may be computed basedonly on a measurement of the I/Q impairments at the frequency f (oralternatively, only on a measurement of the I/Q impairments at thefrequency −f). (C) In another such embodiment, both diagonal componentsof the 2×2 matrix may correspond to pure pass-through filters; one ofthe non-diagonal components may be set to zero; and a frequency responseof the other non-diagonal component at the arbitrary frequency f may becomputed based only on a measurement of the I/Q impairments at thefrequency f (or alternatively, only on a measurement of the I/Qimpairments at the frequency −f).

In another embodiment, a system and method for configuring a receiver toat least partially compensate for I/Q impairments of the receiver mayinvolve the following operations. Measurements of the I/Q impairments ofthe receiver over a frequency band are received (or accessed frommemory). A 2×2 matrix of digital filters is computed based on themeasurements. The 2×2 matrix of digital filters is computed to achieveat least partial compensation for the I/Q impairments of the receiverover the frequency band. The frequency response of at least one of thediagonal components of the 2×2 matrix is computed based on themeasurements as a function of frequency and the measurements as afunction of the negative of frequency. In addition, the frequencyresponse of at least one of the non-diagonal components of the 2×2matrix is computed based on the measurements as a function of frequencyand the measurements as a function of the negative of frequency. Adigital circuit is then programmed to implement the 2×2 matrix ofdigital filters. When so programmed, the digital circuit is configuredto at least partially compensate for the I/Q impairments of the receiverover the frequency band. The digital circuit may be realized in any of awide variety of forms. For example, the digital circuit may be realizedby a programmable hardware element, or by dedicated digital circuitrysuch as an ASIC, or by a processor in response to the execution ofprogram instructions. (The digital circuit may be incorporated as partof the receiver, or as part of another system, e.g., a host computer orcontroller board).

In another embodiment, a system and method for operating a transmitterso as to achieve I/Q impairment compensation may involve the followingoperations.

A digital inphase (I) signal and a digital quadrature (Q) signal arereceived. The digital I signal and the digital Q signal is filtered inaccordance with a 2×2 matrix of digital filters to produce a filtereddigital I signal and a filtered digital Q signal. The 2×2 matrix ofdigital filters at least partially pre-compensates for I/Q impairmentsof the transmitter over a range of frequencies. The frequency responseof at least one of the diagonal components of the 2×2 matrix is computedbased on measurements of the I/Q impairments as a function of frequencyand the measurements as a function of the negative of frequency. (Themeasurements of the transmitter's I/Q impairments may be obtained by anyknown method. The present document describes a number of methods forobtaining such measurements.) Moreover, the frequency response of atleast one of the non-diagonal components of the 2×2 matrix is computedbased on the measurements as a function of frequency and themeasurements as a function of the negative of frequency. The filtereddigital I and Q signals may then be converted to analog form in order toobtain respective analog I and Q signals. I/Q modulation may beperformed on the analog I and Q signals to produce a modulated analogsignal.

In some embodiments, the transmitter's I/Q impairments over positivefrequencies and the transmitter's I/Q impairments over negativefrequencies may be assumed to be functionally related. (A) In one suchembodiment, the computation of the 2×2 matrix of digital filters may besimplified as follows. The frequency response of at least one of thediagonal components of the 2×2 matrix at the arbitrary frequency f inthe frequency range may be computed based only on a measurement of theI/Q impairments at the frequency f (or alternatively, only on ameasurement of the I/Q impairments at the frequency −f). Furthermore,the frequency response of at least one of the non-diagonal components ofthe 2×2 matrix at the frequency f may be computed based only on themeasurement of the I/Q impairments at the frequency f (or alternatively,only on the measurement of the I/Q impairments at the frequency −f). (B)In another such embodiment, the gain imbalance is assumed to be even andthe phase skew is assumed to be odd. Then both non-diagonal componentsof the 2×2 matrix may be set to zero; one of the diagonal components maycorrespond to a pure pass-through filter (i.e., unity frequencyresponse); and a frequency response of the other diagonal component atthe arbitrary frequency f may be computed based only on a measurement ofthe I/Q impairments at the frequency f (or alternatively, only on ameasurement of the I/Q impairments at the frequency −f). (C) In anothersuch embodiment, both diagonal components of the 2×2 matrix maycorrespond to pure pass-through filters; one of the non-diagonalcomponents may be set to zero; and a frequency response of the othernon-diagonal component at the arbitrary frequency f may be computedbased only on a measurement of the I/Q impairments at the frequency f(or alternatively, only on a measurement of the I/Q impairments at thefrequency −f).

In another embodiment, a system and method for configuring a transmitterto at least partially compensate for I/Q impairments of the transmittermay involve the following operations.

Measurements of the I/Q impairments of the transmitter over a frequencyrange are received (or accessed from memory). A 2×2 matrix of digitalfilters is computed based on the measurements. The 2×2 matrix of digitalfilters is computed to achieve at least partial pre-compensation for theI/Q impairments of the transmitter. The frequency response of at leastone of the diagonal components of the 2×2 matrix is computed based onthe measurements as a function of frequency and the measurements as afunction of the negative of frequency. Furthermore, the frequencyresponse of at least one of the non-diagonal components of the 2×2matrix is computed based on the measurements as a function of frequencyand the measurements as a function of the negative of frequency. Adigital circuit is then programmed to implement the 2×2 matrix ofdigital filters. When so programmed, the digital circuit is configuredto at least partially pre-compensate for the I/Q impairments of thetransmitter.

In another embodiment, a system and method for operating a transmitterso as to achieve at least partial compensation for I/Q impairments ofthe transmitter at a given frequency f may involve the followingoperations.

A digital inphase (I) signal and a digital quadrature (Q) signal arereceived. The digital I signal and the digital Q signal are transformedin accordance with a 2×2 matrix of constants to produce a resultantdigital I signal and a resultant digital Q signal. (In other words, thevector signal comprising the digital I signal and digital Q signal ismultiplied by the 2×2 matrix.) The resultant digital I and Q signals areconverted to analog form in order to obtain respective analog I and Qsignals. I/Q modulation is performed on the analog I and Q signals toproduce a modulated analog signal. The 2×2 matrix is configured to atleast partially pre-compensate for the I/Q impairments at frequency f. Afirst of the constants, corresponding a diagonal element of the 2×2matrix, is computed based on a measurement of the I/Q impairments atfrequency f and a measurement of the I/Q impairments at frequency −f.Furthermore, a second of the constants, corresponding to a non-diagonalelement of the 2×2 matrix, is computed based on the measurement atfrequency f and the measurement at frequency −f.

In another embodiment, a method for determining (i.e., measuring) I/Qimpairments of a transmitter may involve the following actions.

The method involves performing a set of operations. The set ofoperations includes: (a) directing that a complex exponential tone atfrequency f be supplied to the transmitter; (b) supplying apre-compensation transformation to a pre-compensation circuit of thetransmitter, where the pre-compensation circuit is configured to applythe pre-compensation transformation to the complex exponential tone toobtain an adjusted complex signal, where the pre-compensationtransformation is configured to pre-compensate for a current estimate ofthe I/Q impairments of the transmitter, where the transmitter isconfigured to transmit a transmit signal based on the adjusted complexsignal, where a receiver is configured to receive the transmit signaland capture a sampled complex signal representing the received transmitsignal; (c) computing raw I/Q impairments based on the sampled complexsignal; (d) transforming the raw I/Q impairments to determinetransformed I/Q impairments, where said transforming removes measuredI/Q impairments of the receiver from the raw I/Q impairments; (e)removing a current estimate of a signal path from the transformed I/Qimpairments to obtain path-compensated I/Q impairments, where the signalpath includes a path from an I/Q modulator of the transmitter to ademodulator of the receiver; and (f) updating the current estimate ofthe I/Q impairments of the transmitter based on the path-compensated I/Qimpairments. (The demodulator may be an I/Q demodulator or not,depending the architecture of the receiver.)

In another embodiment, a method for determining I/Q impairments of atransmitter may involve the following actions.

The method may include configuring a local oscillator (LO) of thetransmitter and a local oscillator (LO) of the receiver to be phaselocked to a common reference and so that a frequency of the receiver'sLO minus a frequency of the transmitter's LO is equal (e.g., exactlyequal) to an amount ΔLO.

The method may also include performing a set of operations, where theset of operations includes: (a) directing that a complex exponentialtone at frequency f be supplied to the transmitter; (b) supplying apre-compensation transformation to a pre-compensation circuit of thetransmitter, where the pre-compensation circuit is configured to applythe pre-compensation transformation to the complex exponential tone inorder to obtain an adjusted complex signal, where the pre-compensationtransformation is configured to pre-compensate for a current estimate ofthe I/Q impairments of the transmitter, where the transmitter isconfigured to transmit a transmit signal based on the adjusted complexsignal, where a receiver is configured to receive the transmit signaland to capture a sampled complex signal representing the receivedtransmit signal; (c) frequency shifting the sampled complex signal bythe amount ΔLO to obtain a frequency-shifted signal; (d) computing rawI/Q impairments at frequency f based on the frequency-shifted signal;(e) removing a current estimate of a signal path from the raw I/Qimpairments at frequency f to obtain path-compensated I/Q impairments atfrequency f, where the signal path includes a path from an I/Q modulatorof the transmitter to a demodulator of the receiver; and (f) updatingthe current estimate of the I/Q impairments of the transmitter atfrequency f based on the path-compensated I/Q impairments at frequencyf. (The demodulator may be an I/Q demodulator or not, depending thearchitecture of the receiver.)

In another embodiment, a method for determining (i.e., measuring) I/Qimpairments of a receiver may involve the following actions.

The method may involve directing that an input signal be supplied to thereceiver, where the input signal includes an isolated tone atdisplacement frequency f and includes a void interval arounddisplacement frequency −f (In one embodiment, the receiver includes acalibration tone generator that is configured to generate the inputsignal.) The receiver is configured to demodulate the input signal inorder to obtain a sampled complex signal. The displacement frequencies fand −f are displacements relative to a local oscillator frequency of thereceiver.

The method may also involve computing the I/Q impairments of thereceiver at frequency f based on the sampled complex signal.

The method may also involve repeating the actions of directing andcomputing for values of the frequency f spanning a specified frequencyband.

The method may also involve storing the I/Q impairments of the receiverfor the values of the frequency f in a memory.

In another embodiment, a method for estimating I/Q impairmentsassociated with a sampled complex signal produced by a receiver mayinvolve the following actions.

A device is directed to stimulate the receiver with a stimulus signalhaving an isolated tone at displacement frequency f and a void intervalat displacement frequency −f (The displacement frequencies f and −f aredisplacements with respect to a local oscillator frequency of thereceiver. The sampled complex signal may be a baseband signal producedby the receiver.) A Discrete-Time Fourier transform value C_(I) atfrequency f is computed for an I component of the sampled complexsignal. A Discrete-Time Fourier transform value C_(Q) at frequency f iscomputed for a Q component of the sampled complex signal. A gainimbalance g of the sampled complex signal at frequency f is computedbased on magnitudes of the values C_(I) and C_(Q). The gain imbalance gincludes at least a gain imbalance of the receiver. A phase skew φ ofthe sampled complex signal at frequency f is computed based on phases ofthe values C_(I) and C_(Q), where the phase skew φ includes at least aphase skew of the receiver.

In another embodiment, a method for estimating a DC scaling of a signalpath between an I/Q modulator of a transmitter and an I/Q demodulator ofa receiver may involve the following operations. To facilitate thisestimation method, the output of the transmitter may be coupled to theinput of the receiver, e.g., via a cable.

The transmitter is directed to supply a zero signal as input to the I/Qmodulator. A first response signal that has been captured from the I/Qdemodulator in response to said supplying the zero signal is received.The transmitter is directed to supply a constant signal equal to anon-zero complex constant as input to the I/Q modulator. A secondresponse signal that has been captured from the I/Q demodulator inresponse to said supplying the constant signal is received. The firstresponse signal is averaged to obtain a first average, and the secondresponse signal is averaged to obtain a second average. A differencebetween the second average and the first average is computed. The DCscaling is computed based on the difference and the non-zero complexconstant. Furthermore, a DC rotation of the signal path may be computedbased on a phase of the difference and a phase of the non-zero complexconstant. The DC scaling and DC rotation are usable to remove an effectof the signal path from measured I/Q impairments at the receiver inorder to obtain estimates of the I/Q impairments of the transmitter.

In one alternative embodiment of the above-described method of DCscaling/rotation estimation, the transmitter has no (or negligible)local oscillator leakage. (Such might be the case, e.g., when thetransmitter has an RF architecture other than a direct conversionarchitecture.) Thus, one may omit the transmission of the zero signal,the capture of the first response signal, the computation of the firstaverage, and the computation of the difference. The DC scaling is thencomputed based on the second average and the non-zero complex constant.The DC rotation is computed based on a phase of the second average andthe phase of the non-zero complex constant.

In another embodiment, a method for computing I/Q impairments at acomplex output (i.e., an I/Q output pair) of an electrical system basedon I/Q impairments at a complex input (i.e., an I/Q input pair) of theelectrical system may include the following operations.

A spectrum A(f) is computed according to the expression

H(f)(1+g(f)exp(jφ(f)),

where H(f) is a spectrum of a linear system model of the electricalsystem, where g(f) is a gain imbalance at the complex input, where φ(f)is a phase skew at the complex input. A spectrum B(f) is computedaccording to the expression

H(−f)(1−g(f)exp(−jφ(f)).

A sum of the spectra A(f) and B(f) and a difference of the spectra A(f)and B(f) are computed. A gain imbalance and phase skew at the complexoutput are computed based on real and imaginary parts of the sum, andreal and imaginary parts of the difference.

In some embodiments, the electrical system being modeled by the spectrumH(f) is the inverse of a signal path from an I/Q modulator of atransmitter to a demodulator of a receiver, e.g., as variously describedherein. The gain imbalance and the phase skew at the complex input ofthe electrical system may represent a gain imbalance and a phase skew atthe input (or alternatively, at the output) of the demodulator. The gainimbalance and the phase skew at the complex output of the electricalsystem may represent a gain imbalance and a phase skew at the output ofthe I/Q modulator.

Various embodiments of communication devices and associated methods forreducing I/Q impairments in signals used by the communication devicesare described herein. According to one embodiment, a receiver device mayreceive a transmission signal over a communication medium, and mayperform I/Q demodulation on the received transmission signal to producea pair of analog I (in-phase) and Q (quadrature) signals. The receiverdevice may perform analog-to-digital conversion of each of the analog Iand Q signals to produce respective digital I and Q signals. Theresulting digital I and Q signals may have I/Q impairments caused by theI/Q demodulation and/or the analog-to-digital conversion and/or otherprocessing. The receiver device may be configured to perform widebandI/Q impairment correction on the digital I and Q signals to correct theI/Q impairments. The wideband I/Q impairment correction may compensatefor frequency-dependent variations of gain imbalance and phase imbalancein the digital I and Q signals, e.g., may compensate for gain imbalancesand phase imbalances in the digital I and Q signals at a plurality offrequency offsets across an instantaneous bandwidth of the receiverdevice.

Performing the wideband I/Q impairment correction on the digital I and Qsignals may comprise filtering one or more of the digital I signal orthe digital Q signal to produce a resultant digital I signal and aresultant digital Q signal. The resultant digital I and Q signalsrepresent corrected signals. In some embodiments, the resultant digitalI signal is identical to the digital I signal, and the resultant digitalQ signal is generated by filtering one or more of the digital I signaland the digital Q signal to obtain one or more respective filteredsignals and by adding the one or more filtered signals. In otherembodiments, the resultant digital Q signal is identical to the digitalQ signal, and the resultant digital I signal is generated by filteringone or more of the digital I signal and the digital Q signal to obtainone or more respective filtered signals and by adding the one or morefiltered signals. In yet other embodiments, the resultant digital Isignal is generated by filtering one or more of the digital I signal andthe digital Q signal to obtain respectively one or more filtered signalsand by adding the one or more filtered signals; and the resultantdigital Q signal is generated by filtering one or more of the digital Isignal and the digital Q signal to obtain respectively one or moreadditional filtered signals and by adding the one or more additionalfiltered signals.

In further embodiments a calibration system (or the receiver deviceitself) may determine correction information by providing a plurality ofknown test signals to the receiver device and measuring I/Q impairmentsintroduced by the receiver device in response to the known test signals.(In one embodiment, the receiver device may include a calibration tonegenerator to generate the known test signal.) The wideband I/Qimpairment correction may utilize the correction information tocompensate for the frequency-dependent variations of gain imbalance andphase imbalance in the digital I and Q signals.

In some embodiments the calibration system may operate in an offlinecalibration phase and an online operation phase. Performing the offlinecalibration phase may include providing a plurality of known testsignals to the receiver device, measuring I/Q impairments introduced bythe receiver device in response to the known test signals, anddetermining correction information based on the measured I/Qimpairments. Performing the online operation phase may include receivinga transmission signal over a communication medium, performing I/Qdemodulation on the received transmission signal to produce analog I andQ signals, performing analog-to-digital conversion of each of the analogI and Q signals to produce digital I and Q signals, and performingwideband I/Q impairment correction on the digital I and Q signals. Thewideband I/Q impairment correction may use the correction informationdetermined in the offline calibration phase to compensate forfrequency-dependent variations of gain imbalance and phase imbalance inthe digital I and Q signals.

In some embodiments the offline calibration phase may be performed inresponse to the receiver device being powered on. In some embodimentsthe receiver device may automatically enter the online operation phasein response to determining that the offline calibration phase iscomplete. In some embodiments the receiver device may automaticallyswitch from the online operation phase to the offline calibration phasein response to determining that the receiver device is not busyprocessing received transmission signals in the online operation phase.In some embodiments the offline calibration phase may be initiated inresponse to user input.

According to other embodiments, a transmitter device may receive digitalI (in-phase) and Q (quadrature) signals to be transmitted. Thetransmitter device may perform wideband I/Q impairment pre-correction onthe digital I and Q signals. The action of performing the wideband I/Qimpairment pre-correction may involve filtering one or more of thedigital I and Q signals to produce a resultant digital I signal and aresultant digital Q signal to pre-compensate for frequency-dependentvariations of gain imbalance and phase imbalance that will besubsequently introduced during synthesis of a transmission signal. Thetransmission signal may be synthesized using the resultant digital Isignal and the resultant digital Q signal.

The action of synthesizing the transmission signal may includeperforming digital-to-analog conversion of the resultant digital Isignal and the resultant digital Q signal to produce an analog I signaland an analog Q signal, and performing I/Q modulation to produce thetransmission signal using the analog I signal and the analog Q signal.The resultant digital I signal and the resultant digital Q signal maypre-compensate for frequency-dependent variations of gain imbalance andphase imbalance caused by one or more of the digital to analogconversion and the I/Q modulation.

In some embodiments, the resultant digital I signal is identical to thedigital I signal, and the resultant digital Q signal is generated byfiltering one or more of the digital I signal and the digital Q signalto obtain one or more respective filtered signals and by adding the oneor more filtered signals. In other embodiments, the resultant digital Qsignal is identical to the digital Q signal, and the resultant digital Isignal is generated by filtering one or more of the digital I signal andthe digital Q signal to obtain one or more respective filtered signalsand by adding the one or more filtered signals. In yet otherembodiments, the resultant digital I signal is generated by filteringone or more of the digital I signal and the digital Q signal to obtainrespectively one or more filtered signals and by adding the one or morefiltered signals; and the resultant digital Q signal is generated byfiltering one or more of the digital I signal and the digital Q signalto obtain respectively one or more additional filtered signals and byadding the one or more additional filtered signals.

In further embodiments, a calibration system may determine correctioninformation by providing a plurality of known digital test signals tothe transmitter device and measuring I/Q impairments introduced by thetransmitter device in response to the known test signals. The widebandI/Q impairment pre-correction may utilize the correction information toproduce the resultant digital signals.

In some embodiments the transmitter device may operate in an offlinecalibration phase and an online operation phase. The offline calibrationphase may include providing a plurality of known test signals to thetransmitter device, measuring I/Q impairments introduced by thetransmitter device in response to the known test signals, anddetermining correction information based on the measured I/Qimpairments.

In some embodiments the offline calibration phase may be performed inresponse to the transmitter device being powered on. In some embodimentsthe transmitter device may automatically enter the online operationphase in response to determining that the offline calibration phase iscomplete. In some embodiments the transmitter device may automaticallyswitch from the online operation phase to the offline calibration phasein response to determining that the transmitter device is not busytransmitting signals in the online operation phase. In some embodimentsthe offline calibration phase may be initiated in response to userinput.

The online operation phase may include receiving digital I and Q signalsto be transmitted, and performing wideband I/Q impairment pre-correctionon the digital I and Q signals. The action of performing the widebandI/Q impairment pre-correction may use the correction informationdetermined in the offline calibration phase to filter one or more of thedigital I and Q signals to produce a resultant digital I signal and aresultant digital Q signal to pre-compensate for frequency-dependentvariations of gain imbalance and phase imbalance that will besubsequently introduced during synthesis of a transmission signal. Thetransmission signal may be synthesized using the resultant digital Isignal and a resultant digital Q signal

According to another embodiment, a measurement system may include areceiver device and a device under test. The receiver device may beconfigured to receive a transmission signal including measurement dataacquired from the device under test, perform I/Q demodulation on thereceived transmission signal to produce analog I (in-phase) and Q(quadrature) signals, perform analog-to-digital conversion of each ofthe analog I and Q signals to produce digital I and Q signals, andperform wideband I/Q impairment correction on the digital I and Qsignals. The wideband I/Q impairment correction may compensate forfrequency-dependent variations of gain imbalance and phase imbalance inthe digital I and Q signals.

In further embodiments, the measurement system may also include atransmitter device. The transmitter device may be configured to receivedigital I and Q signals to be transmitted. The digital I and Q signalsmay specify information to be transmitted to the device under test. Thetransmitter device may be further configured to perform wideband I/Qimpairment pre-correction on the digital I and Q signals. The action ofperforming the wideband I/Q impairment pre-correction may involvefiltering one or more of the digital I and Q signals to produce aresultant digital I signal and a resultant digital Q signal topre-compensate for frequency-dependent variations of gain imbalance andphase imbalance that will be subsequently introduced during synthesis ofa transmission signal. The transmitter device may synthesize thetransmission signal using the resultant digital I signal and a resultantdigital Q signal, and transmit the transmission signal to the deviceunder test.

BRIEF DESCRIPTION OF THE DRAWINGS

A better understanding of the present inventions can be obtained whenthe following detailed description is considered in conjunction with thefollowing drawings.

FIG. 1A illustrates one possible application of the compensation methodsdisclosed herein, where mobile device 10 and/or wireless transceiverstation 15 apply digital pre-compensation to their transmitted signalsand/or digital post-compensation to their received signals.

FIG. 1B illustrates another possible application of the compensationmethods disclosed herein, where a test instrument 20 applies digitalpre-compensation to remove the effect of its I/Q impairments on thesignals it transmits to a receiver under test 25.

FIG. 1C illustrates yet another possible application of the compensationmethods disclosed herein, where a test instrument 35 applies digitalpost-compensation to remove the effect of its I/Q impairments on thesignals it receives from a transmitter under test.

FIG. 2A illustrates one embodiment of a method for operating a receiverso as to achieve at least partial I/Q impairment compensation.

FIG. 2B illustrates one embodiment of a receiver configured to achieveat least partial I/Q impairment compensation.

FIG. 3 illustrates one embodiment of a method for configuring a receiverto enable the receiver to at least partially compensate for I/Qimpairments.

FIG. 4 illustrates one embodiment of a method for operating atransmitter so as to achieve at least partial I/Q impairmentcompensation.

FIG. 5 illustrates one embodiment of a transmitter configured to achieveat least partial I/Q impairment compensation.

FIG. 6 illustrates one embodiment of a method for configuring atransmitter to enable the transmitter to at least partially compensatefor I/Q impairments.

FIG. 7 illustrates one embodiment of a system configured to provide I/Qimpairment compensation. The I/Q impairments are modeled as appearingentirely on the Q channel.

FIG. 8 illustrates another embodiment of a system configured to provideI/Q impairment compensation. The I/Q impairments are modeled asappearing entirely on the I channel.

FIG. 9 illustrates yet another embodiment of a system configured toprovide I/Q impairment compensation. The I/Q impairments are modeled asappearing partially on both channels.

FIG. 10 illustrates one embodiment of a method for operating a receiverso as to achieve at least partial compensation for I/Q impairments atfrequency f.

FIG. 11 illustrates one embodiment of a receiver that is configured toachieve at least partial compensation for I/Q impairments at frequencyf.

FIG. 12 illustrates one embodiment of a method for configuring areceiver to enable the receiver to achieve at least partial compensationfor I/Q impairments at frequency f.

FIG. 13 illustrates one embodiment of a method for operating atransmitter so as to achieve at least partial compensation for I/Qimpairments at frequency f.

FIG. 14 illustrates one embodiment of a transmitter that is configuredto achieve at least partial compensation for I/Q impairments atfrequency f.

FIG. 15 illustrates a system being stimulated by a complex exponentialtone and a distorted complex exponential tone appearing at the systemoutput, where the distortions is characterized by a gain imbalance andphase skew.

FIG. 16 illustrates a system where the gain imbalance and phase skewappear entirely on the Q channel.

FIG. 17 illustrates one embodiment of a system for performing impairmentcompensation at a single frequency.

FIG. 18 illustrates a 2×2 system model for performing I/Q impairmentcompensation.

FIG. 19 illustrates an embodiment where an impairment model G precedesthe compensation model H.

FIG. 20A illustrates an embodiment where the impairment model G followsthe compensation model H.

FIG. 20B illustrates the response of the same series combination (ofmodel H followed by model G) to a complex exponential tone at frequencyf.

FIG. 21 illustrates one embodiment for the compensation model H in termsof a pair of digital filters having frequency responses U(f) and V(f)respectively.

FIG. 22 illustrates a refinement of FIG. 21, where U and V arerepresented in terms of their even and odd parts.

FIG. 23 illustrates an equivalent representation for the system of FIG.22, where the odd spectra B and D are replaced by corresponding evenspectra followed by Hilbert Transforms.

FIGS. 24A and 24B illustrate the responses of the system FIG. 23 to tworespective inputs.

FIG. 25 presents equations that are derived from FIGS. 24A and 24Brespectively.

FIGS. 26A and 26B illustrate phasor diagrams corresponding to theequations of FIG. 25.

FIG. 27 presents equations according to one embodiment that specify thecompensation spectra A, E_(B), C and E_(D) in terms of the I/Qimpairment information.

FIG. 28 illustrates a 2×2 model H that represents the I/Q impairments ofa system.

FIG. 29 illustrates one embodiment of the model H, in terms of frequencyresponses U and V.

FIG. 30 illustrates a refinement of FIG. 29, where U and V arerepresented in terms of their even and odd parts.

FIG. 31 illustrates an equivalent representation for the system of FIG.30, where the odd spectra B and D are replaced by corresponding evenspectra followed by Hilbert Transforms.

FIGS. 32A and 32B illustrate the responses of the system FIG. 31 to tworespective inputs.

FIG. 33 presents equations that are derived from FIGS. 32A and 32Brespectively.

FIGS. 34A and 34B illustrate phasor diagrams corresponding to theequations of FIG. 33.

FIG. 35 presents a matrix equation derived from the phasor diagrams ofFIGS. 34A and 34B.

FIG. 36 presents the solution to the matrix equation of FIG. 35.

FIG. 37 illustrates one embodiment of a system for measuring propertiesof a signal path between the I/Q modulator 3710 and the I/Q demodulator3735.

FIG. 38 illustrates a LO leakage vector A, an intentionally-injected DCvector B and their sum C.

FIG. 39 illustrates the response vectors A′, B′ and C′ correspondingrespectively to vectors A, B and C.

FIG. 40 illustrates one embodiment of a method for computing a DCmapping value for the signal path.

FIG. 41 illustrates a system with frequency response H(f) beingstimulated by an input signal s_(input)(f,t) having gain imbalance g(f)and phase skew φ(f), and producing output signal s_(output)(f,t) withgain imbalance g′(f) and phase skew φ′(f).

FIG. 42 presents equations derived from FIG. 41.

FIG. 43 illustrates one embodiment of a method for transforming I/Qimpairments through a linear system H(f).

FIG. 44 illustrates one embodiment of a method for determining I/Qimpairments of a transmitter.

FIG. 45 illustrates one embodiment of a method for determining I/Qimpairments of a transmitter, using intentionally-displaced localoscillators.

FIG. 46 illustrates one embodiment of a method for determining the I/Qimpairments of a receiver.

FIG. 47 illustrates one embodiment of a method for estimating I/Qimpairments associated with a complex signal.

FIG. 48 illustrates one embodiment of a system for measuring transmitterand/or receiver I/Q impairments, where the system includes a transmitterand receiver whose local oscillator frequencies are intentionallyoffset.

FIG. 49 illustrates the spectrum of the signal received by the receiverin response to the transmitter's transmission of a tone at 31 MHz. Thetransmitter's local oscillator frequency is 6 MHz higher than thereceiver's local oscillator frequency. Thus, the tone appears at 37 MHzin the received spectrum.

FIG. 50 illustrates the received spectrum after removal of thereceiver's I/Q impairments.

FIG. 51 illustrates the spectrum of FIG. 50 after having been frequencyshifted.

FIG. 52 illustrates the frequency-shifted spectrum without firstremoving the receiver's impairments.

FIG. 53A illustrates a single-point vector calibration correction 5310followed by a double-point vector corruption model 5320.

FIG. 53B shows a refinement of FIG. 53A, where the single-point vectorcalibration correction is determined by constants α and β, and where thedouble-point vector corruption is determined by constants A, E_(B), Cand E_(D).

FIG. 54 illustrates a phasor diagram corresponding to the right-handpart of FIG. 53B (i.e., to the right of the dotted line).

FIG. 55A illustrates a receiver including a receiver filter 5525 and anI/Q demodulator 5530.

FIG. 55B illustrates a system including a transmitter and a receiverthat are coupled together. The system may be used to determine the I/Qimpairments of the transmitter and/or the receiver.

FIG. 55C illustrates the relative magnitude of the tone at frequency fand the image at −f at three points along the path from thetransmitter's I/Q modulator to the receiver's I/Q demodulator.

FIG. 56A illustrates convergence rate as a function of magnitudeestimation error.

FIG. 56B illustrates convergence rate as a function of rotation (phase)estimation error.

FIG. 57 introduces notation for the complex amplitude a of a tone andthe complex amplitude β of an image carried by complex signal that hasbeen distorted by gain imbalance g(f) and phase skew φ(f).

FIGS. 58A and 58B derive equations that characterize the tone and imagein terms of the gain imbalance g(f) and phase skew φ(f).

FIG. 59 illustrates the gain imbalance g(f) and phase skew φ(f) in termsof a distortion to the Q channel signal (“Q Actual”) relative to the Ichannel signal (“I Reference”).

FIGS. 60 and 61 show the magnitude spectra for the in-phase andquadrature signal components, i.e., for the “I Reference” signal and “Qactual” signal of FIG. 59.

FIG. 62 illustrates a LabVIEW graphical program for computing localoscillator leakage, signal amplitude, gain imbalance, image rejectionand phase skew, according to one embodiment.

FIG. 63 illustrates shows a LabVIEW graphical program (VI) that receivesdata computed by a programmable hardware element (e.g., an FPGA of areceiver), and computes LO Leakage, amplitude gain imbalance and phaseskew from that data.

FIGS. 64 and 65 show plots of the amplitude spectrum of a rectangularwindow function with different acquisition lengths and with a commonsample rate of 120 MHz.

FIG. 66 illustrates a system model whose complex input signal has I/Qimpairments g_(in)(ω) and φ_(in)(ω) and whose complex output signal hasI/Q impairments g_(out)(ω) and φ_(out)(ω).

FIG. 67 presents equations specifying the frequency response functionsU(ω) and V(ω) of FIG. 66 in terms of the input I/Q impairments g_(in)(ω)and φ_(in)(ω) and the output I/Q impairments g_(out)(ω) and φ_(out)(ω).

FIG. 68 illustrates one embodiment of a computer system 6800 that may beused to perform any of the method embodiments described herein.

While the invention is susceptible to various modifications andalternative forms, specific embodiments thereof are shown by way ofexample in the drawings and are herein described in detail. It should beunderstood, however, that the drawings and detailed description theretoare not intended to limit the invention to the particular formdisclosed, but on the contrary, the intention is to cover allmodifications, equivalents and alternatives falling within the spiritand scope of the present invention as defined by the appended claims.Note that the various section headings in the following DetailedDescription are for organizational purposes only and are not meant to beused to limit the claims.

DETAILED DESCRIPTION Terminology

The following is a glossary of terms used in the present application:

Memory Medium—Any of various types of memory devices or storage devices.The term “memory medium” is intended to include an installation medium,e.g., a CD-ROM, floppy disks 105, or tape device; a computer systemmemory or random access memory such as DRAM, DDR RAM, SRAM, EDO RAM,Rambus RAM, etc.; a non-volatile memory such as a Flash, magnetic media,e.g., a hard drive, or optical storage; registers, or other similartypes of memory elements, etc. The memory medium may comprise othertypes of memory as well or combinations thereof. In addition, the memorymedium may be located in a first computer in which the programs areexecuted, or may be located in a second different computer whichconnects to the first computer over a network, such as the Internet. Inthe latter instance, the second computer may provide programinstructions to the first computer for execution. The term “memorymedium” may include two or more memory mediums which may reside indifferent locations, e.g., in different computers that are connectedover a network.

Programmable Hardware Element—includes various hardware devicescomprising multiple programmable function blocks connected via aprogrammable interconnect. Examples include FPGAs (Field ProgrammableGate Arrays), PLDs (Programmable Logic Devices), FPOAs (FieldProgrammable Object Arrays), and CPLDs (Complex PLDs). The programmablefunction blocks may range from fine grained (combinatorial logic or lookup tables) to coarse grained (arithmetic logic units or processorcores). A programmable hardware element may also be referred to as“reconfigurable logic”.

Computer System—any of various types of computing or processing systems,including a personal computer system (PC), mainframe computer system,workstation, network appliance, Internet appliance, personal digitalassistant (PDA), television system, grid computing system, or otherdevice or combinations of devices. In general, the term “computersystem” can be broadly defined to encompass any device (or combinationof devices) having at least one processor that executes instructionsfrom a memory medium.

Local Oscillator (LO)—a circuit configured to generate a periodic signalat a specified frequency and amplitude. The periodic signal may be apure sinusoid, and its frequency and/or amplitude may be programmable.The periodic signal may or may not be phase or frequency locked toanother periodic signal.”

Embodiments of the present invention may be realized in any of variousforms. For example, in some embodiments, the present invention may berealized as a computer-implemented method, a computer-readable memorymedium, or a computer system. In other embodiments, the presentinvention may be realized using one or more custom-designed hardwaredevices such as ASICs. In other embodiments, the present invention maybe realized using one or more programmable hardware elements such asFPGAs.

In some embodiments, a computer-readable memory medium may be configuredso that it stores program instructions and/or data, where the programinstructions, if executed by a computer system, cause the computersystem to perform a method, e.g., any of a method embodiments describedherein, or, any combination of the method embodiments described herein,or, any subset of any of the method embodiments described herein, or,any combination of such subsets.

In some embodiments, a computer system may be configured to include aprocessor (or a set of processors) and a memory medium, where the memorymedium stores program instructions, where the processor is configured toread and execute the program instructions from the memory medium, wherethe program instructions are executable to implement any of the variousmethod embodiments described herein (or, any combination of the methodembodiments described herein, or, any subset of any of the methodembodiments described herein, or, any combination of such subsets). Thecomputer system may be realized in any of various forms. For example,the computer system may be a personal computer (in any of its variousrealizations), a workstation, a computer on a card, anapplication-specific computer in a box, a server computer, a clientcomputer, a hand-held device, a tablet computer, a wearable computer,etc.

In some embodiments, a set of computers distributed across a network maybe configured to partition the effort of executing a computationalmethod (e.g., any of the method embodiments disclosed herein). In someembodiments, a first computer may be configured to receive an O-QPSKmodulated signal and to capture samples of that signal. The firstcomputer may send the samples to a second computer through the network.The second computer may operate on the samples according to any of themethod embodiments described herein, or, any combination of the methodembodiments described herein, or, any subset of any of the methodembodiments described herein, or, any combination of such subsets.

FIG. 1A illustrates one possible application (among many) of theinventive ideas described herein. A mobile device 10 (e.g., mobilephone) communicates wirelessly with a wireless transceiver station 15.The mobile device 10 may include the digital pre-correction describedherein to improve the quality of its transmitted signals, i.e., tocorrect for so-called “I/Q impairments” in its transmission hardware(e.g., in its I/Q modulator). Similarly, the wireless transceiverstation 15 may apply digital post-correction to its received signal tocorrect for I/Q impairments in its reception hardware (e.g., in its I/Qdemodulator). Furthermore, the wireless transceiver station and mobiledevice may apply the same pre-correction and post-correction with rolesexchanged, i.e., for transmissions in the opposite direction.

FIG. 1B illustrates another possible application of the inventive ideasdescribed herein. A test transmitter 20 transmits signals to a receiverunder test 25. The test transmitter 20 may perform the digitalpre-correction described herein correct for its own I/Q impairments, andthus, improve the quality of its transmissions. For example, the testtransmitter 20 may achieve a higher standard on image rejection due touse of the digital pre-correction. Thus, distortions (e.g., I/Qimpairments) measured in the signals captured by the receiver may beascribed to the receiver's imperfections.

FIG. 1C illustrates yet another possible application of the inventiveideas described herein. A test receiver 35 receives signals transmittedby a transmitter under test 30. The test receiver employs the digitalpost-correction described herein to correct for its own I/Q impairments.Thus, the receiver may meet a higher standard on image rejection than itwould without the post-correction. Therefore, any distortions (e.g., I/Qimpairments) measured in the signals captured by the receiver may beclearly assigned to the transmitter's imperfections.

Wideband Correction Method for Receiver

In one set of embodiments, a method 100 for compensating for I/Qimpairments of a receiver over a range of frequencies may involve theoperations shown in FIG. 2A.

At 110, the receiver may receive an analog input signal. The analoginput signal may be received from a transmission medium. Thetransmission medium is a medium that permits the transmission of signalenergy. For example, the transmission medium may be free space, theatmosphere, the earth or some portion of the earth's surface, anelectrical cable, a fiber optic cable, a body of water such as an ocean.

At 115, the receiver may perform I/Q demodulation on the analog inputsignal to produce an analog inphase (I) signal and an analog quadrature(Q) signal. The process of I/Q demodulation is well understood in thefield of communication. Typically, I/Q demodulation involves mixing theanalog input signal with a pair of orthogonal carriers. For example, themixing may be interpreted according to the following model:

I(t)=y(t)cos(ωt)

Q(t)=y(t)sin(ωt).

In some embodiments, the analog I signal and the analog Q signals may beinterpreted as baseband signals, i.e., as the components of a complexbaseband signal. In other embodiments, the analog I signal and theanalog Q signals may be interpreted as intermediate frequency (IF)signals.

At 120, the receiver may digitize the analog I signal and the analog Qsignal to produce, respectively, a digital I signal and a digital Qsignal. (The term “digital signal” is meant to imply a sampled signal,not a two-state signal.) Thus, the receiver may include a pair ofanalog-to-digital converters (ADCs).

At 125, the digital I signal and the digital Q signal may be filtered inaccordance with a 2×2 matrix of digital filters to produce a filtereddigital I signal and a filtered digital Q signal. The filtering mayinvolve applying the 2×2 matrix (h_(ij)) of digital filters according tothe relations:

I _(F)(n)=h ₁₁(n)*I(n)+h ₁₂(n)*Q(n)

Q _(F)(n)=h ₂₁(n)*I(n)+h ₂₂(n)*Q(n),

where the symbol “*” represents convolution. (Note, elsewhere in thispatent disclosure, the symbol “*” may mean convolution ormultiplication, depending on the particular context. As a superscript,“*” denotes complex conjugation.)

The 2×2 matrix of digital filters may compensate (or, at least partiallycompensate) for I/Q impairments of the receiver over a range offrequencies, e.g., a range of frequencies wide enough to cover thebandwidth of transmitted communication signal or an instantaneousbandwidth of the receiver. (Processes for measuring I/Q impairments arediscussed at length later in this patent disclosure.) In other words,the digital filters make the receiver's input-output behavior moreclosely approximate a perfect receiver that has no I/Q impairments. Inresponse to the application of a pure sinusoidal tone at arbitraryfrequency ω as input, a perfect receiver would produce signals I(n) andQ(n) that are equal in amplitude and 90 degrees apart in phase, i.e., nogain imbalance and no phase skew.

The 2×2 matrix of digital filters may have the following properties. Thefrequency response of at least one of the diagonal components of the 2×2matrix may be computed based on measurements of the I/Q impairments as afunction of frequency and the measurements of the I/Q impairments as afunction of the negative of frequency. For example, if one characterizesthe I/Q impairments with a gain imbalance function g(f) and a phase skewfunction φ(f), with f covering the range of frequencies, the frequencyresponse of the component h₂₂ (or the component h₁₁, or each of thecomponents h₁₁ and h₂₂) may be computed based on the functions g(f),g(−f), φ(f) and φ(−f).

Furthermore, the frequency response of at least one of the non-diagonalcomponents of the 2×2 matrix may be computed based on the measurementsof the I/Q impairments as a function of frequency and the measurementsof the I/Q impairments as a function of the negative of frequency.

By saying that the filtering of the digital I signal and the digital Qsignal is performed “in accordance with a 2×2 matrix of digital filters”is not meant to suggest that the receiver (or whatever device is used toimplement the filtering) must include a filter circuit to implement atrivial multiplication by zero when the corresponding element of the 2×2matrix is identically zero, or an adder to implement a trivial additionby zero. As an example, if h₁₂=0, then I_(F)(n) may be computedaccording to the simplified expression

I _(F)(n)=h ₁₁(n)*I(n)

with only one convolution circuit. Similarly, if one of the componentsof the 2×2 matrix is a unit impulse at time n=0, then the receiver neednot include a multiplier to perform that trivial convolution. Forexample, if h₁₁(n) is a unit impulse, then I_(F)(n) may be simplycomputed according to the expression

I _(F)(n)=I(n)+h ₁₂(n)*Q(n)

with only one convolution unit and one adder. Thus, filtering “inaccordance with a 2×2 matrix of digital filters” does not necessarilyrequire a full 2×2 array of convolution circuits in all cases.

In some embodiments, the filtered digital I signal and the filtereddigital Q signal are usable to recover a stream of information bits. Thereceiver (or another processing agent such as a host computer) mayrecover the stream of information bits by performing symbol demodulationon the filtered digital I signal and the filtered digital Q signal. Insymbol demodulation, the vector signal (I_(F)(n),Q_(F)(n)) may bedecimated to determine a sequence of complex symbols, and each of thecomplex symbols may be mapped to the closest constellation point in agiven constellation (set of points in the complex plane). The sequenceof resulting complex points determines a stream of information bits.

In some embodiments, the receiver includes a digitizer, where thedigitizer performs the above-described actions of digitizing andfiltering. The term “digitizer” is meant to imply an instrument that iscalibrated to a known standard. For example, the relationship betweenthe analog input and the digital output is calibrated to a knownstandard for both the I channel and the Q channel.

In some embodiments, the receiver is a test instrument such as a vectorsignal analyzer (VSA). (The term “vector signal” is a synonym forcomplex signal or I/Q signal.) The test instrument may receive theanalog input signal from a transmitter, e.g., a transmitter under test.The analog input signal is received in response to the transmitter'saction of transmitting a transmit signal onto the transmission medium.The test instrument may be configured to compensate for its own I/Qimpairments, but to not compensate for the I/Q impairments of thetransmitter. In the context of test and measurement, it is important tobe able to accurately measure and report the impairments of a deviceunder test rather than to compensate for the impairments of that device.Thus, for a test instrument, it may be preferable that the measurementsof the receiver's I/Q impairments (on which the receiver's impairmentcompensation is based) do not include I/Q impairments of thetransmitter. The present patent disclosure describes methods formeasuring receiver-only impairments.

Test instruments are generally used to perform the testing of devicesunder test (DUTs) or systems under test (SUTs). Test instrumentsgenerally include one or more inputs and outputs for connecting to SUTs.The inputs and outputs may be analog, digital, radio frequency, etc.,e.g., at various voltage levels and frequencies. Test instruments aregenerally able to perform one or more tests or features. For example,test instruments may be configured to capture and analyze waveforms,calculate measured power, generate a tone at a programmed frequency,etc. Test instruments are also typically calibrated in order to achievea specified level of accuracy on its I/O. Finally, test instrumentsusually include a user interface in order to specify how the testinstrument should behave.

In other contexts, the receiver may be expected to compensate for thetransmitter's impairments and its own impairments. Thus, the 2×2 matrixof digital filters may be computed based on measurements of the I/Qimpairments of the transmitter-and-receiver combination. The sameprinciple regarding calculation of frequency responses based on theimpairments as a function of f and the impairments as a function −fapplies here.

In some embodiments, the filtering operation 125 may be performed on aprogrammable hardware element such as an FPGA, or in dedicated digitalcircuitry such as an application specific integrated circuit (ASIC). Theprogrammable hardware element or dedicated digital circuitry may besupplied with the same sample clock that drives the ADC conversion.

In some embodiments, the filtering operation 125 may be performed by aprocessor in response to the execution of program instructions. Theprocessor may be incorporated as part of the receiver, or as part ofanother system such as a host computer or controller board.

As described above, at least one of the diagonal components of the 2×2matrix is computed based on the measurements of the I/Q impairments as afunction of f and the measurements of the impairments as a function −f.In some embodiments, the “at least one diagonal” is to be interpreted as“exactly one diagonal”, and the other diagonal component of the 2×2matrix is a discrete-time unit impulse function (e.g., taking the valueone at time zero, and zero elsewhere).

As described above, at least one of the non-diagonal components of the2×2 matrix is computed based on the measurements of the I/Q impairmentsas a function of f and the measurements of the I/Q impairments as afunction −f. In some embodiments, the “at least one non-diagonal” is tobe interpreted as “exactly one non-diagonal”, and the other non-diagonalcomponent of the 2×2 matrix is the zero function.

Constraint Between Receiver Impairments at Frequency f and Frequency −f

In some embodiments, the receiver's I/Q impairments over positivefrequencies and the receiver's I/Q impairments over negative frequenciesmay be assumed to be functionally related. In one such embodiment, thecomputation of the 2×2 matrix of digital filters may be simplified asfollows. The frequency response of one of the diagonal components of the2×2 matrix at the arbitrary frequency f in the frequency range may becomputed based only on a measurement of the I/Q impairments at thefrequency f (or alternatively, only on a measurement of the I/Qimpairments at the frequency −f). For example, if the I/Q impairmentsare characterized by a gain imbalance function g(f) and a phase skewfunction φ(f), the frequency response H₂₂(f) of the component h₂₂ may becomputed based only on a measurement of g(f) and a measurement of φ(f),where f ranges over the frequencies at which measurements have beenobtained. Furthermore, the frequency response of one of the non-diagonalcomponents of the 2×2 matrix at the frequency f may be computed basedonly on the measurement of the I/Q impairments at the frequency f (oralternatively, only on the measurement of the I/Q impairments at thefrequency −f).

In some embodiments, the I/Q impairments at the frequency f and the I/Qimpairments at frequency −f are constrained such that the I/Qimpairments at f are determined by the I/Q impairments at −f, or suchthat the I/Q impairments at frequency −f are determined by the I/Qimpairments at f. For example, the gain imbalance at the frequency f andthe gain imbalance at frequency −f may be constrained to be equal, andthe phase skew at frequency f and the phase skew at frequency −f may beconstrained to be equal (or negatives of each other).

In some embodiments, the gain imbalance is assumed to be even and thephase skew is assumed to be odd. In these embodiments, both non-diagonalcomponents of the 2×2 matrix may be set to zero; one of the diagonalcomponents may correspond to a pure pass-through filter (i.e., unityfrequency response); and a frequency response of the other diagonalcomponent at the arbitrary frequency f may be computed based only on ameasurement of the I/Q impairments at the frequency f (or alternatively,only on a measurement of the I/Q impairments at the frequency −f).

In some embodiments, both diagonal components of the 2×2 matrix maycorrespond to pure pass-through filters; one of the non-diagonalcomponents may be set to zero; and a frequency response of the othernon-diagonal component at the arbitrary frequency f may be computedbased only on a measurement of the I/Q impairments at the frequency f(or alternatively, only on a measurement of the I/Q impairments at thefrequency −f).

Receiver Configured for Wideband Correction

In one set of embodiments, a receiver 200 may be configured as shown inFIG. 2B. (Receiver 200 may include any subset of the features describedabove in connection with method 100.) Receiver 200 may include an I/Qdemodulator 210, a digitization unit 215 and a digital circuit 220.

The I/Q demodulator 210 may be configured to receive an analog inputsignal y(t) and perform I/Q demodulation on the analog input signal toproduce an analog inphase (I) signal and an analog quadrature (Q)signal, denoted I(t) and Q(t). The I/Q demodulator may receive a pair oforthogonal carriers from a local oscillator circuit.

The digitization unit 215 may be configured to digitize the analog Isignal and the analog Q signal to produce, respectively, a digital Isignal and a digital Q signal, which are denoted I(n) and Q(n). Thedigitization unit 215 may receive a conversion clock from a clockgeneration circuit. The digitization unit includes an I-channel ADC anda Q-channel ADC, each being driven by the same conversion clock.

The digital circuit 220 may be configured to filter the digital I signaland the digital Q signal in accordance with a 2×2 matrix of digitalfilters (as described above) to produce a filtered digital I signal anda filtered digital Q signal. The 2×2 matrix of digital filters may beconfigured to compensate (or, at least partially compensate) for I/Qimpairments of the receiver over a range of frequencies. The digitalcircuit, when programmed with the 2×2 matrix of digital filters, makesthe receiver 200 behave more like a mathematically perfect receiver,i.e., one having a perfect I/Q demodulator and perfect digitizationunit.

In some embodiments, the digital circuit 220 is realized by (or, as partof) a programmable hardware element, or dedicated digital circuitry suchas an ASIC.

In some embodiments, the digital circuit 220 is (or includes, or isrealized by) a processor that is configured to execute programinstructions. In one embodiment, the processor is part of a computersystem such as a host computer system or controller board.

In some embodiments, the receiver 200 may include a means for recoveringa stream of information bits by performing symbol demodulation on thefiltered digital I signal and the filtered digital Q signal. Therecovering means may include any one or more of the following: aprocessor executing on the receiver, a processor executing on a hostcomputer, a processor executing on a controller board (e.g., acontroller board installed in an instrumentation chassis along with thereceiver), a programmable hardware element, an ASIC.

In some embodiments, the receiver 200 is (or includes) a testinstrument. See the above discussion of the notion of a test instrument.

Method for Configuring a Receiver to Perform Impairment Correction

In one set of embodiments, a method 300 for configuring a receiver mayinvolve the operations shown in FIG. 3. The method 300 may be used toconfigure the receiver to at least partially compensate for I/Qimpairments of the receiver. The method 300 may be implemented by acomputer system in response to the execution of program instructions.(The method 300 may include any subset of the features described above.)

At 310, the computer system may receive measurements of the I/Qimpairments of the receiver over a frequency band. (“Over a frequencyband” means that the measurements include measurements at a plurality ofdifferent frequencies within the frequency band, e.g., uniformly ornon-uniformly covering the frequency band.) The receiver may include anI/Q demodulator, a pair of analog-to-digital converters (ADCs) and adigital circuit, e.g., as described above. The I/Q demodulator may beconfigured to generate an analog I signal and an analog Q signal from ananalog input signal. The ADCs may be configured to sample the analog Isignal and the analog Q signal to respectively obtain a digital I signaland a digital Q signal. The digital circuit may be configured to filterthe digital I signal and the digital Q signal to obtain a filtereddigital I signal and a filtered digital Q signal. (See the abovediscussion for various ways of realizing the digital circuit.)

At 315, the computer system may compute a 2×2 matrix of digital filtersbased on the measurements. The 2×2 matrix of digital filters may becomputed to achieve at least partial compensation for the I/Qimpairments of the receiver over the frequency band. A frequencyresponse of at least one of the diagonal components of the 2×2 matrixmay be computed based on the measurements as a function of frequency andthe measurements as a function of the negative of frequency.Furthermore, a frequency response of at least one of the non-diagonalcomponents of the 2×2 matrix may be computed based on the measurementsas a function of frequency and the measurements as a function of thenegative of frequency.

At 320, the computer system may program the digital circuit to implementthe 2×2 matrix of digital filters, where the digital circuit, when soprogrammed, is configured to at least partially compensate for the I/Qimpairments of the receiver over the frequency band. The action ofprogramming the digital circuit involves transferring the 2×2 matrix ofdigital filters (or parameters specifying those filters) to the digitalcircuit or to a memory used by the digital circuit.

Wideband Correction Method for Transmitter

In one set of embodiments, a method 400 for compensating for I/Qimpairments of a transmitter may involve the operations shown in FIG. 4.

At 410, a digital inphase (I) signal and a digital quadrature (Q) signalmay be received. The digital I signal and the digital Q signal may beinterpreted as the components of a complex-valued signal I(n)+jQ(n). Thedigital I signal and the digital Q signal may carry one or more streamsof information bits, e.g., as the result of symbol modulation accordingto a given constellation. In some embodiments, the digital I signal andthe digital Q signal may be interpreted as the components of acomplex-valued baseband signal or intermediate frequency (IF) signal.

At 415, the digital I signal and the digital Q signal may be filtered inaccordance with a 2×2 matrix of digital filters to produce a filtereddigital I signal and a filtered digital Q signal. (The filteringoperation may be performed by the transmitter or some other agent.) Thefiltering operation may involve applying the 2×2 matrix (h_(ij)) ofdigital filters according to the relations:

I _(F)(n)=h ₁₁(n)*I(n)+h ₁₂(n)*Q(n),

Q _(F)(n)=h ₂₁(n)*I(n)+h ₂₂(n)*Q(n).

The 2×2 matrix of digital filters may pre-compensate (or, at leastpartially pre-compensate) for the I/Q impairments of the transmitterover a range of frequencies, e.g., over a frequency range broad enoughto cover the bandwidth of a communication signal to be transmitted.

The 2×2 matrix of digital filters may have the following properties. Thefrequency response of at least one of the diagonal components of the 2×2matrix may be computed based on measurements of the I/Q impairments as afunction of frequency and the measurements of the I/Q impairments as afunction of the negative of frequency. For example, if the I/Qimpairments are characterized by a gain imbalance function g(f) and aphase skew function φ(f), with f covering the range of frequencies, thefrequency response of the digital filter h₂₂ (or the digital filter h₁₁,or each of the digital filters h₁₁ and h₂₂) may be computed based ong(f), g(−f), φ(f) and φ(−f).

Furthermore, a frequency response of at least one of the non-diagonalcomponents of the 2×2 matrix may be computed based on the measurementsof the I/Q impairments as a function of frequency and the measurementsof the I/Q impairments as a function of the negative of frequency.

In the description of the receiver 100, we were careful to qualify themeaning of filtering “in accordance with a 2×2 matrix of digitalfilters”. Those same qualifications apply here for the transmittercompensation.

At 420, the transmitter may convert the filtered digital I signal andthe filtered digital Q signal to analog form in order to respectivelyobtain an analog I signal and an analog Q signal.

At 425, the transmitter may perform I/Q modulation on the analog I and Qsignals to produce a modulated analog signal. The modulated analogsignal may be transmitted onto a transmission medium, e.g., atransmission medium as described above. A receiver may receive themodulated analog signal, likely in a noise-perturbed andchannel-distorted form.

Above, we described the 2×2 matrix of digital filters as“pre-compensating” for I/Q impairments of the transmitter. That isbecause the I/Q impairments occur after the application of the digitalfilters, especially in the I/Q modulation stage. Thus, the 2×2 matrixmay be interpreted as applying an inverse distortion that together withthe following distortions will give an approximation to the identity mapoverall.

In some embodiments, the filtering operation 415 may be performed in aprogrammable hardware element (PHE) such as an FPGA, or in dedicateddigital circuitry such as an application-specific integrated circuit(ASIC).

In some embodiments, the filtering operation 415 may be performed by aprocessor in response to the execution of program instructions, e.g., aprocessor of a host computer system or an instrumentation controllerbroad.

In some embodiments, the transmitter is a test instrument (e.g., anarbitrary waveform generator or a vector signal generator). The testinstrument may transmit the modulated analog signal to a receiver, e.g.,a receiver under test. In the context of test and measurement, it may beimportant for the test instrument to correct for its own impairments butto not correct for the impairments of the receiver. Thus, in thiscontext, the above-described measurements of the transmitter's I/Qimpairments (on which the transmitter's pre-compensation is based)preferably do not include I/Q impairments of the receiver. This patentdisclosure describes methods for measuring transmitter-only impairments(cleanly separated from receiver impairments).

In some contexts, the transmitter may be expected to correct for thereceiver's impairment and its own impairments. Thus, the 2×2 matrix ofdigital filters may be computed based on measurements of the I/Qimpairments of the transmitter-and-receiver combination. The sameprinciple regarding calculation of frequency responses based on theimpairments as a function off and the impairments as a function −fapplies here.

Constraint Between Transmitter Impairments at Frequency f and Frequency−f

In some embodiments, the transmitter's I/Q impairments over positivefrequencies and the transmitter's I/Q impairments over negativefrequencies may be assumed to be functionally related. In one suchembodiment, the computation of the 2×2 matrix of digital filters may besimplified as follows. The frequency response of at least one of thediagonal components of the 2×2 matrix at the arbitrary frequency f inthe frequency range may be computed based only on a measurement of theI/Q impairments at the frequency f (or alternatively, only on ameasurement of the I/Q impairments at the frequency −f). For example, ifthe I/Q impairments are characterized by a gain imbalance function g(f)and a phase skew function φ(f), the frequency response value H₂₂(f) ofthe component h₂₂ may be computed based only on the a measurement ofg(f) and a measurement of φ(f), where f ranges over the frequencies atwhich measurements have been obtained. Furthermore, the frequencyresponse of at least one of the non-diagonal components of the 2×2matrix at the frequency f may be computed based only on the measurementof the I/Q impairments at the frequency f (or alternatively, only on themeasurement of the I/Q impairments at the frequency −f).

In some embodiments, the I/Q impairments at the frequency f and the I/Qimpairments at frequency −f are constrained such that the I/Qimpairments at f are determined by the I/Q impairments at −f, or suchthat the I/Q impairments at frequency −f are determined by the I/Qimpairments at f. For example, the gain imbalance at the frequency f andthe gain imbalance at frequency −f may be constrained to be equal, andthe phase skew at frequency f and the phase skew at frequency −f may beconstrained to be equal (or alternatively, negatives of each other).

In some embodiments, the gain imbalance is assumed to be even and thephase skew is assumed to be odd. Then both non-diagonal components ofthe 2×2 matrix may be set to zero; one of the diagonal components maycorrespond to a pure pass-through filter (i.e., unity frequencyresponse); and a frequency response of the other diagonal component atthe arbitrary frequency f may be computed based only on a measurement ofthe I/Q impairments at the frequency f (or alternatively, only on ameasurement of the I/Q impairments at the frequency −f).

In some embodiments, both diagonal components of the 2×2 matrix maycorrespond to pure pass-through filters; one of the non-diagonalcomponents may be set to zero; and a frequency response of the othernon-diagonal component at the arbitrary frequency f may be computedbased only on a measurement of the I/Q impairments at the frequency f(or alternatively, only on a measurement of the I/Q impairments at thefrequency −f).

Transmitter Configured for Wideband Correction

In one set of embodiments, a transmitter 500 may be configured as shownin FIG. 5. (Transmitter 500 may incorporate any subset of the featuresdescribed above in connection with method 400.) Transmitter 500 mayinclude a digital circuit 510, a digital-to-analog conversion (DAC) unit515 and an I/Q modulator 520.

The digital circuit 510 may be configured to receive a digital inphase(I) signal and a digital quadrature (Q) signal, and filter the digital Isignal and the digital Q signal with a 2×2 matrix of digital filters toproduce a filtered digital I signal and a filtered digital Q signal.(The filtering may be performed as variously described above.) Thedigital I signal and the digital Q signal may carry one or more streamsof information bits.

The 2×2 matrix of digital filters may be computed to pre-compensate (or,at least partially pre-compensate) for I/Q impairments of thetransmitter over a range of frequencies. A frequency response of atleast one of the diagonal components of the 2×2 matrix may be computedbased on measurements of the I/Q impairments as a function of frequencyand the measurements of the I/Q impairments as a function of thenegative of frequency. Furthermore, a frequency response of at least oneof the non-diagonal components of the 2×2 matrix may be computed basedon the measurements of the I/Q impairments as a function of frequencyand the measurements of the I/Q impairments as a function of thenegative of frequency.

The digital circuit 510 is said to “pre-compensate” for the I/Qimpairment of the transmitter because the I/Q impairments occur intransmitter stages after the digital circuit, especially in the I/Qmodulator 520. Thus, the digital circuit (by applying the 2×2 matrix ofdigital filters) introduces a pre-distortion to the complex signalI(n)+jQ(n) so that the net effect of the pre-distortion followed by thesubsequent impairments will approximate an ideal transmitter having noI/Q impairments. In other words, the digital circuit applies an inversedistortion that in combination with the subject distortion approximatesthe identity map (i.e., frequency response function identically equal tounity).

The DAC unit 515 may be configured to convert the filtered digital I andQ signals to analog form in order to obtain respective analog I and Qsignals. The DAC unit 515 may receive a conversion clock from a clockgeneration unit. The digital circuit 510 may receive the same conversionclock so that it generates the complex samples (I_(F)(n),Q_(F)(n)) atthe same rate that the DAC unit converts the samples into analog form(I(t),Q(t)).

The I/Q modulator 520 may be configured to perform I/Q modulation on theanalog I and Q signals to produce a modulated analog signal. Themodulated analog signal may be transmitted to a receiver through atransmission medium. The notion of I/Q modulation is well understood inthe field of communication. For example, the I/Q modulation may bemodeled by the expressions:

$\begin{matrix}{{x(t)} = {{{I(t)}{\cos \left( {\omega \; t} \right)}} - {{Q(t)}{\sin \left( {\omega \; t} \right)}}}} \\{{= {{Re}\left\{ {\left( {{I(t)} + {j\; {Q(t)}}} \right){\exp \left( {j\; \omega \; t} \right)}} \right\}}},}\end{matrix}$

where ω is the carrier frequency.

In some embodiments, the digital circuit 510 is realized by (or, as partof) a programmable hardware element, or dedicated digital circuitry suchas an ASIC.

In some embodiments, the digital circuit 510 is (or includes, or isrealized by) a processor that is configured to execute programinstructions. In one embodiment, the processor is part of a computersystem such as a host computer system or controller board.

In some embodiments, the transmitter 500 may be a test instrument. Seethe above discussion of test instrument in the context of method 400.

Method for Configuring a Transmitter for Impairment Correction

In one set of embodiments, a method 600 for configuring a transmittermay involve the operations shown in FIG. 6. The method 600 may be usedto configure the transmitter to at least partially compensate for I/Qimpairments of (or introduced by) the transmitter. The method 600 may beperformed by a computer system in response to the execution of programinstructions.

At 610, the computer system may receive measurements of the I/Qimpairments of the transmitter over a frequency range. (“Over afrequency range” implies that the measurements of the I/Q impairmentsare obtained at a plurality of frequencies within the frequency range,e.g., frequencies covering the frequency range uniformly ornon-uniformly.) The transmitter may include a digital circuit, a pair ofdigital-to-analog converters (DACs) and an I/Q modulator. The digitalcircuit may be configured to filter a digital I signal and a digital Qsignal to respectively obtain a filtered digital I signal and a filtereddigital Q signal. The pair of DACs may be configured to convert thefiltered digital I signal and the filtered digital Q signal to analogform in order to respectively obtain an analog I signal and an analog Qsignal. The I/Q modulator may be configured to modulate a carrier signalwith the analog I and Q signals to obtain a modulated carrier signal.The modulated carrier signal may be transmitted to a receiver through atransmission channel.

At 615, the computer system may compute a 2×2 matrix of digital filtersfor the digital circuit based on the measurements. The 2×2 matrix ofdigital filters may be computed to achieve pre-compensation (or, atleast partial pre-compensation) for the I/Q impairments of thetransmitter over the frequency range. The frequency response of at leastone of the diagonal components of the 2×2 matrix may be computed basedon the measurements as a function of frequency and the measurements as afunction of the negative of frequency. Furthermore, the frequencyresponse of at least one of the non-diagonal components of the 2×2matrix may be computed based on the measurements as a function offrequency and the measurements as a function of the negative offrequency.

At 620, the computer system may program the digital circuit to implementthe 2×2 matrix of digital filters, where the digital circuit, when soprogrammed, is configured to at least partially pre-compensate for theI/Q impairments over the frequency range. The action of programming thedigital circuit involves transferring the digital filters (or parametersspecifying the filters) to the digital circuit or to parameter memoryused by the digital circuit.

In various embodiments, the digital circuit may be a programmablehardware element, an application specific integrated circuit (ASIC), aprocessor executing under the control of program instructions, or anycombination thereof.

Derivation of Digital Filters for Wideband Impairment Compensation

As described above, a 2×2 matrix of digital filters may be used tocompensate for I/Q impairments at a receiver or a transmitter. (Indeed,both the transmitter and the receiver may employ the matrixcompensation, each using its own 2×2 compensation matrix. Thetransmitter's compensation matrix may be computed based on thetransmitter's I/Q impairments, while the receiver's compensation matrixmay be computed based on the receiver's I/Q impairments.) This sectionderives the frequency responses for the digital filters in the specialcase where the 2×2 matrix has the special form shown in FIG. 7.

Since gain imbalance g and phase skew φ are relative measurements, wehave the freedom to model the gain imbalance and phase skew as being dueto distortions on only one channel (I or Q), the other channel beingideal. FIG. 7 represents the choice of modeling both the gain imbalanceand the phase skew as being due to distortions on the Q channel only.FIG. 8 illustrates the opposite choice. (Thus, frequency responses H₁₁and H₁₂ are used to effect the compensation, while H₂₂=1 and H₂₁=0.) Onemight also model the gain imbalance as being due to amplitude distortionon one channel only, and the phase skew as being due to phase distortionon the opposite channel only. As yet another alternative, one mightmodel the gain imbalance and/or phase skew as being due to partialdistortions on both channel, e.g., as suggested by FIG. 9. Thus, thedigital compensation may be performed using all four frequency responsesH₁₁, H₁₂, H₂₁ and H₂₂. After appreciating the following derivation basedon FIG. 7, one of ordinary skill in the art will find it straightforwardto apply the same mathematical principles to all other cases.

FIG. 7 may be interpreted as the filtering operation performed by thereceiver's digital circuit 220 or the filtering operation performed bythe transmitter's digital circuit 510. Thus, the following derivationapplies both to the transmitter's compensation matrix and to thereceiver's compensation matrix.

While the compensation is applied digitally, for the sake of simplicity,the following derivation will be expressed in terms of continuous timet. To achieve compensation, we seek frequency responses U(ω) and V(ω) sothat the distorted signal

cos(ωt)+jg(ω)sin(ωt+φ(ω))

gets transformed to corrected signal cos(ωt)+j sin(ωt) for allfrequencies ω in a frequency band (e.g., a frequency band that issymmetric about zero), or at least at selected frequencies wheremeasurements of the impairments g(ω) and φ(ω) are available. g(ω) is thegain imbalance corresponding to frequency ω, and φ(ω) is the phase skewcorresponding to frequency ω. Thus, we obtain the equation:

u(t)*cos(ωt)+v(t)*g(ω)sin(ωt+φ(ω))=sin(ωt),

where “*” denotes convolution, where u(t) and v(t) are the impulseresponses corresponding respectively to the frequency responses U(ω) andV(ω).

By making the substitutions

cos(θ)=(½){exp(jθ)+exp(−jθ)}

sin(θ)=(−j/2){exp(jθ)−exp(−jθ)},

we obtain the equation

(½)U(ω)exp(jωt)+(½)U(−ω)exp(jωt)+(−j/2)V(ω)g(ω)exp(jφ(ω))exp(jωt)+(j/2)V(−ω)g(ω)exp(−jφ(ω))exp(jωt)=(−j/2)[exp(jωt)−exp(−jωt)}.

Due to the linear independence of exp(jωt) and exp(jωt), we obtain thefollowing two equations:

jU(ω)+V(ω)g(ω)exp(jφ(ω))=1  (a):

jU(−ω)−V(−ω)g(ω)exp(−jφ(ω))=−1.  (b):

Because equation (b) holds for all ω, we can replace ω with −ω, thusobtaining equation (b′) below.

jU(ω)−V(ω)g(−ω)exp(−jφ(−ω))=−1.  (b′):

Equations (a) and (b′) specify a matrix equation in vector unknown[U(ω),V(ω)]^(T), whose solution is given by:

${{U(\omega)} = {j\; \frac{{{g(\omega)}\exp \left\{ {j\; {\phi (\omega)}} \right\}} - {{g\left( {- \omega} \right)}{\exp\left( {{- j}\; {\phi \left( {- \omega} \right)}} \right\}}}}{{{g(\omega)}\exp \left\{ {j\; {\phi (\omega)}} \right\}} + {{g\left( {- \omega} \right)}{\exp\left( {{- j}\; {\phi \left( {- \omega} \right)}} \right\}}}}}},{{V(\omega)} = {\frac{2}{{{g(\omega)}{\exp\left( {j\; {\phi (\omega)}} \right\}}} + {{g\left( {- \omega} \right)}\exp \left\{ {{- j}\; {\phi \left( {- \omega} \right)}} \right\}}}.}}$

Observe that U(ω) and V(ω) each depend on g(ω), g(−ω), φ(ω) and φ(−ω).This property of the frequency responses (of the digital filters)depending on impairment information at ω and −ω applies more generallythan to the special matrix form of FIG. 7. Indeed, it applies to anyform of the compensation matrix. Also observe that U and V are conjugatesymmetric with respect to frequency: U(−ω)=U(ω)* and V(−ω)=V(ω)*, asexpected for filters whose impulses responses are entirely real.

To simplify the process of designing digital filters (impulse responses)corresponding to the frequency responses U(ω) and V(ω), it may be usefulto express those frequency responses in terms of their even and oddparts:

U(ω)=A(ω)+B(ω)

A(ω)=(½){U(ω)+U(−ω)}

B(ω)=(½){U(ω)−U(−ω)}

V(ω)=C(ω)+D(ω)

C(ω)=(½){V(ω)+V(−ω)}

D(ω)=(½){V(ω)−V(−ω)}.

In the time domain, the corresponding expressions are:

u(t)=a(t)+b(t)

a(t)=(½){u(t)+u(−t)}

b(t)=(½){u(t)−u(−t)}

v(t)=c(t)+d(t)

c(t)=(½){v(t)+v(−t)}

d(t)=(½){v(t)−v(−t)},

where u, a, b, v, c, and d are the impulse responses correspondingrespectively to frequency responses U, A, B, V, C and D.

Using the above-derived expressions for U(ω) and V(ω), it follows that:

${A(\omega)} = \frac{{- 2}{g(\omega)}{g\left( {- \omega} \right)}\sin \left\{ {{\phi (\omega)} + {\phi \left( {- \omega} \right)}} \right\}}{{g^{2}(\omega)} + {g^{2}\left( {- \omega} \right)} + {2{g(\omega)}{g\left( {- \omega} \right)}\cos \left\{ {{\phi (\omega)} + {\phi \left( {- \omega} \right)}} \right\}}}$${B(\omega)} = \frac{j\left\{ {{g^{2}(\omega)} - {g^{2}\left( {- \omega} \right)}} \right\}}{{g^{2}(\omega)} + {g^{2}\left( {- \omega} \right)} + {2{g(\omega)}{g\left( {- \omega} \right)}\cos \left\{ {{\phi (\omega)} + {\phi \left( {- \omega} \right)}} \right\}}}$${C(\omega)} = \frac{{2{g(\omega)}\cos \left\{ {\phi (\omega)} \right\}} + {2{g\left( {- \omega} \right)}\cos \left\{ {\phi (\omega)} \right\}}}{{g^{2}(\omega)} + {g^{2}\left( {- \omega} \right)} + {2{g(\omega)}{g\left( {- \omega} \right)}\cos \left\{ {{\phi (\omega)} + {\phi \left( {- \omega} \right)}} \right\}}}$${D(\omega)} = {\frac{2j\left\{ {{{- {g(\omega)}}{\sin \left( {\phi (\omega)} \right)}} + {{g\left( {- \omega} \right)}{\sin \left( {\phi \left( {- \omega} \right)} \right)}}} \right\}}{{g^{2}(\omega)} + {g^{2}\left( {- \omega} \right)} + {2{g(\omega)}{g\left( {- \omega} \right)}\cos \left\{ {{\phi (\omega)} + {\phi \left( {- \omega} \right)}} \right\}}}.}$

The above expressions may be used to compute frequency responses U and Vbased on measured or estimated impairment functions g and φ. Theseexpressions apply equally to post-compensation at the receiver orpre-compensation at the transmitter. In other words, the frequencyresponses U(ω) and V(ω) for pre-correcting I/Q impairments g(f) and ω(f)are the same as the frequency responses for post-correction those sameI/Q impairments.

The computed frequency responses U and V may be used to determinecorresponding impulses responses u(n) and v(n) using any of variousknown filter design algorithms.

Note Regarding Filters with Odd Frequency Response

Given a filter with odd frequency response B(ω), it is basic fact thatthe function E_(B)(ω) given by

E _(B)(ω)=jB(ω)sgn(ω)

is even and has the property that:

b(t)*x(t)=HT(e _(B)(t)*x(t)),

where HT is the Hilbert transform operator, where b(t) is the impulseresponse corresponding to B(ω), and x(t) is an arbitrary input function,where sgn(ω) is 1 if ω is greater than zero and −1 if ω is less thanzero.

If we apply this fact to the odd functions B(ω) and D(ω) from thediscussion above, we arrive at the corresponding even functions:

${E_{B}(\omega)} = \frac{\left\{ {{g^{2}\left( {- \omega} \right)} - {g^{2}(\omega)}} \right\} {{sgn}(\omega)}}{{g^{2}(\omega)} + {g^{2}\left( {- \omega} \right)} + {2{g(\omega)}{g\left( {- \omega} \right)}\cos \left\{ {{\phi (\omega)} + {\phi \left( {- \omega} \right)}} \right\}}}$${E_{D}(\omega)} = {\frac{2{{sgn}(\omega)}\left\{ {{{g(\omega)}{\sin \left( {\phi (\omega)} \right)}} - {{g\left( {- \omega} \right)}{\sin \left( {\phi \left( {- \omega} \right)} \right)}}} \right\}}{{g^{2}(\omega)} + {g^{2}\left( {- \omega} \right)} + {2{g(\omega)}{g\left( {- \omega} \right)}\cos \left\{ {{\phi (\omega)} + {\phi \left( {- \omega} \right)}} \right\}}}.}$

Note Regarding Special Case of Even g(ω) and Odd φ(ω)

In many circumstances, the gain imbalance function may be modeled asbeing even and the phase skew function may be modeled as being odd,i.e., g(ω)=g(−ω) and φ(ω)=−φ(−ω). Under these constraints, U(ω)=0 andV(ω) is complex.

Note Regarding Special Case of Even g(ω) and Even φ(ω)

The above-derived expressions for U(ω) and V(ω) are typically complexvalued. However, when the gain imbalance and phase skew functions areeven, i.e., g(ω)=g(−ω) and φ(ω)=φ(−ω), it follows that U(ω) and V(ω)become real valued:

U(ω)=−tan(φ(ω))

V(ω)=1/{g(ω)cos φ(ω)}.

Constant Matrix to Post-Correct Receiver Impairments at a SingleFrequency

In one set of embodiments, a method 1000 for operating a receiver (oroperating a system including a receiver) may involve the operationsshown in FIG. 10.

At 1010, the receiver may receive an analog input signal. The analoginput signal may be received from a transmission medium, e.g., asdescribed above.

At 1015, the receiver may perform I/Q demodulation on the analog inputsignal to produce an analog inphase (I) signal and an analog quadrature(Q) signal, e.g., as described above.

At 1020, the receiver may digitize the analog I signal and the analog Qsignal to produce respectively a digital I signal and a digital Qsignal.

At 1025, the receiver may transform the digital I signal and the digitalQ signal in accordance with a 2×2 matrix c=(c_(ij)) of constants toproduce a resultant digital I signal and a resultant digital Q signal.The transformation may be performed by applying the following matrixmultiplication:

${\begin{bmatrix}{I_{R}(n)} \\{Q_{R}(n)}\end{bmatrix} = {\begin{bmatrix}c_{11} & c_{12} \\c_{21} & c_{22}\end{bmatrix}\begin{bmatrix}{I(n)} \\{Q(n)}\end{bmatrix}}},$

where I_(R)(n) and Q_(R)(n) represent the resultant digital I signal andthe resultant digital Q signal respectively. The 2×2 matrix c may beconfigured to post-compensate (or, at least partially post-compensate)for measured I/Q impairments of the receiver at a particular frequencyf.

The matrix c may have the following properties. At least one of thediagonal elements c₁₁ and c₂₂ may be computed based on the measured I/Qimpairments of the receiver at frequency f. For example, the coefficientc₂₂ may be computed as a function of the measured value g(f) and/or themeasured value φ(f), where g is the gain imbalance function and φ is thephase skew function. Similarly, at least one of the non-diagonalelements c₁₂ and c₂₁ may be computed based on the measured I/Qimpairments of the receiver at frequency f. For example, the coefficientc₂₁ may be computed as a function of the measured value g(f) and/or themeasured value φ(f). In some embodiments, each of the four matrixelements is similarly computed (i.e., based on the measured impairmentsat f).

See the section “Performing Traditional Impairment Compensation at aSingle Frequency” for one possible embodiment of the matrix c.

Let c_(ij)(f) denote the functional expression that is used to determinethe coefficient c_(ij) from the I/Q impairments at frequency f. Due tothe continuity of the functional expressions c_(ij)(f) with respectfrequency f, the matrix c(f) is a good approximation for the matrixc(f+Δf) as long as Δf is sufficiently small. Thus, when the receiveruses the matrix c(f) to perform transform operation 1025, the receiverwill achieve at least partial compensation over a neighborhood offrequencies around f. The quality of the compensation will typicallydegrade as Δf increases in absolute value.

In some embodiments, the analog input signal is a pure sinusoidal tone,e.g., a tone at frequency f or a tone at frequency f+f_(LO), wheref_(LO) is the frequency of the receiver's local oscillator. In otherembodiments, the analog input signal is a communication signal thatcarries a stream of binary information.

In some embodiments, the matrix c has the additional property that oneof its diagonal elements is one. In some embodiments, the matrix c hasthe additional property that one of its non-diagonal elements is zero.In some embodiments, the matrix c has one of the following specialforms:

${c = \begin{bmatrix}1 & 0 \\c_{21} & c_{22}\end{bmatrix}},{c = {\begin{bmatrix}c_{11} & {\; c_{12}} \\0 & 1\end{bmatrix}.}}$

As described above, the transform operation 1025 is performed “inaccordance with a 2×2 matrix”. That qualifying phrase is not meant torequire that the receiver include a multiplier (or adder) to implement atrivial multiplication by one (or a trivial addition by zero). Forexample, in the first special form given above, the resultant digital Isignal is equal to the digital I signal: I_(R)(n)=I(n). This requires nocomputation at all. One can simply pass the I(n) input to the I_(R)(n)output.

In one set of embodiments, a receiver 1100 may be configured as shown inFIG. 11. (The receiver 1100 may incorporate any subset of the featuresdescribed above in connection with method 1000.) The receiver 1100 mayinclude an I/Q demodulator 1110, a digitization unit 1115 and a digitalcircuit 1120.

The I/Q demodulator 1110 may be configured to receive an analog inputsignal, and perform I/Q demodulation on the analog input signal toproduce an analog inphase (I) signal and an analog quadrature (Q)signal. The analog input signal may be received from a transmissionmedium as described above.

The digitization unit 1115 may be configured to digitize the analog Isignal and the analog Q signal to produce respectively a digital Isignal and a digital Q signal.

The digital circuit 1120 may be configured to transform the digital Isignal and the digital Q signal in accordance with a 2×2 matrix ofconstants to produce a resultant digital I signal and a resultantdigital Q signal. The 2×2 matrix may be configured to at least partiallycompensate for I/Q impairments of the receiver at a particular frequencyf. A first of the constants, corresponding to a first diagonal elementof the 2×2 matrix, may be computed based on measured I/Q impairments ofthe receiver at frequency f. Furthermore, a second of the constants,corresponding to a first non-diagonal element of the 2×2 matrix, may becomputed based on the measured I/Q impairments at frequency f. In someembodiments, each of the four constants is similarly computed (i.e.,based on the measured impairments at f).

In one set of embodiments, a method 1200 for configuring a receiver mayinvolve the operations shown in FIG. 12. The method 1200 may be used toconfigure the receiver to at least partially compensate for I/Qimpairments of the receiver at a given frequency f. The method 1200 maybe implemented by a computer system in response to the execution ofprogram instructions. (The method 1200 may include any subset of thefeatures described above in connection with FIGS. 10 and 11.)

At 1210, the computer system may receive measured I/Q impairments of thereceiver at frequency f. The receiver may include an I/Q demodulator, ananalog-to-digital conversion (ADC) unit and a digital circuit, e.g., asdescribed above in connection with FIGS. 10 and 11. The I/Q demodulatormay be configured to generate an analog I signal and an analog Q signalfrom an analog input signal. The ADC unit may be configured to samplethe analog I signal and the analog Q signal to respectively obtain adigital I signal and a digital Q signal. The digital circuit may beconfigured to transform the digital I signal and the digital Q signal toobtain a resultant digital I signal and a resultant digital Q signal.(See the discussions above for various ways of realizing the digitalcircuit.)

At 1215, the computer system may compute a 2×2 matrix of constants basedon the measured I/Q impairments at frequency f. The 2×2 matrix may becomputed to achieve at least partial compensation for the measured I/Qimpairments at frequency f. At least one of the diagonal components ofthe 2×2 matrix may be computed based on the measured I/Q impairments atfrequency f. Furthermore, at least one of the non-diagonal components ofthe 2×2 matrix may be computed based on the measured I/Q impairments atfrequency f.

At 1220, the computer system may program the digital circuit toimplement the 2×2 matrix of constants, where the digital circuit, whenso programmed, is configured to at least partially compensate for themeasured I/Q impairments at frequency f. The action of programming thedigital circuit involves transferring the 2×2 matrix (or informationspecifying the matrix) to the digital circuit or to a memory used by thedigital circuit.

True Matrix Pre-Correction at a Single Frequency

In one set of embodiments, a method 1300 for compensating for the I/Qimpairments of a transmitter at a particular frequency f may involve theoperations shown in FIG. 13.

At 1310, a digital inphase (I) signal and a digital quadrature (Q)signal (e.g., as variously described above) may be received. In someembodiments, the digital I signal and the digital Q signal together mayrepresent a complex exponential tone at the frequency f. In otherembodiments, the digital I signal and the digital Q signal may carryrespective streams of binary information. The digital I and Q signalsmay be the components of a complex baseband signal or of a complexintermediate-frequency signal.

At 1315, the digital I signal and the digital Q signal may betransformed in accordance with a 2×2 matrix c=(c_(ij)) of constants toproduce a resultant digital I signal and a resultant digital Q signal.(The transformation may be performed by the transmitter or some otheragent.) The transformation may be described by the following matrixmultiplication:

${\begin{bmatrix}{I_{R}(n)} \\{Q_{R}(n)}\end{bmatrix} = {\begin{bmatrix}c_{11} & c_{12} \\c_{21} & c_{22}\end{bmatrix}\begin{bmatrix}{I(n)} \\{Q(n)}\end{bmatrix}}},$

where I_(R)(n) and Q_(R)(n) represent the resultant digital I signal andthe resultant digital Q signal respectively. The 2×2 matrix may beconfigured to pre-compensate (or, at least partially pre-compensate) forthe I/Q impairments of the transmitter at the frequency f. See the abovediscussion regarding the nature of “pre-compensation”. In brief, theapplication of the transform introduces an inverse distortion that, incombination with the distortions of the following transmitter stages,makes the transmitter appear more ideal in its input-output behavior.)Note that the above discussion regarding the meaning of transforming “inaccordance with a 2×2 matrix” applies here as well.

The 2×2 matrix c may have the following properties. At least one of thediagonal elements c₁₁ and c₂₂ may be computed based on a measurement ofthe I/Q impairments at frequency f and a measurement of the I/Qimpairments at frequency −f. For example, the diagonal element c₂₂ maybe computed based on a measured value of each of g(f), g(−f), φ(f) andφ(−f), where g is the gain imbalance function and φ is the phase skewfunction. For example, the diagonal element c₂₂ may be computed based ona measured value of each of g(f), g(−f), φ(f) and φ(−f), where g is thegain imbalance function and φ is the phase skew function. Furthermore,at least one of the non-diagonal elements c₁₂ and c₂₁ may be computedbased on the measurement at frequency f and the measurement at frequency−f. In some embodiments, each of the four coefficients may be computedbased on the measurement at frequency f and the measurement at frequency−f See the section “Calculating True Single Point Vector CalibrationConstants” for one possible embodiment of the matrix c.

The measured impairments may be impairments measured at the output ofthe corruption (i.e. the I/Q modulator) and may be different than theimpairments if they could be measured at the input. Alternatively, themethod may include transforming the output impairments at +f and −f toinput impairments at only +f and then computing the matrix constantsaccording to simplified formulas using the input impairments at only +f.The transformation may be derived follows. First, derive specializedexpressions for U(f) and V(f) based on the output impairments at +f and−f using equations (7.9) and (7.10) with g_(in)(f)=g_(in)(−f)=1 andφ_(in)(f)=φ_(in)(−f)=0. Then compute the input impairments g_(in)(f) andφ_(in)(f) based on equation (7.7) with g_(out)(f)=1 and φ_(out)(f)=0:

g _(in)(f)exp(jφ _(in)(f))={1−jU(f)}/V(f).

The matrix constants may then be determined based on g_(in)(f) andφ_(in)(f), e.g., according to the relations α=g_(in)(f)sin(φ_(in)(f))and β=g_(in)(f)cos(φ_(in)(f)).

The quality of the compensation achieved by operation 1315 will belimited by the quality of the impairment measurements. The presentpatent disclosure describes methods for obtaining quality measurementsof the transmitter's I/Q impairments at any given frequency, or, over awhole range of frequencies.

Let c_(ij)(f) denote the functional expression that is used to determinethe coefficient c_(ij) from the I/Q impairments at frequency f and theI/Q impairments at frequency −f. Due to the continuity of the functionalexpressions c_(ij)(f) with respect frequency f, the matrix c(f) is agood approximation for the matrix c(f+Δf) as long as Δf is sufficientlysmall. Thus, when the transmitter uses the matrix c(f) to performtransform operation 1315, the transmitter will achieve at least partialcompensation over a neighborhood of frequencies around f. The quality ofthe compensation will typically degrade as Δf increases in absolutevalue.

At 1320, the transmitter may convert the resultant digital I and Qsignals to analog form in order to obtain respective analog I and Qsignals.

At 1325, the transmitter may perform I/Q modulation on the analog I andQ signals to produce a modulated analog signal, e.g., as describedabove.

In some embodiments, the matrix c has one of the following specialforms:

${c = \begin{bmatrix}1 & 0 \\c_{21} & c_{22}\end{bmatrix}},{c = {\begin{bmatrix}c_{11} & {\; c_{12}} \\0 & 1\end{bmatrix}.}}$

In the first special form above, the constants c₂₁ and c₂₂ may becomputed based on the values A(f), E_(B)(f), C(f) and E_(D)(f) asdescribed in the section “Calculating True Single Point VectorCalibration Constants”, especially at equations (1.81) and (1.82).

In some embodiments, the transformation 1315 may be performed on aprogrammable hardware element such as an FPGA, or in dedicated digitalcircuitry such as an application specific integrated circuit (ASIC). Theprogrammable hardware element or dedicated digital circuitry may besupplied with the same sample clock that drives the ADC conversion.

In some embodiments, the transformation 1315 may be performed by aprocessor in response to the execution of program instructions. Theprocessor may be incorporated as part of the transmitter, or as part ofanother system such as a host computer or controller board.

In one set of embodiments, a transmitter 1400 may be configured as shownFIG. 14. (Transmitter 1400 may include any subset of the featuresdescribed above in connection with method 1300.) Transmitter 1400 mayinclude a digital circuit 1410, a DAC unit 1415 and an I/Q modulator1420.

The digital circuit 1410 may be configured to receive a digital inphase(I) signal and a digital quadrature (Q) signal, and transform thedigital I signal and the digital Q signal in accordance with a 2×2matrix of constants to produce a resultant digital I signal and aresultant digital Q signal. The digital circuit 1410 may be realized inany of various forms, e.g., as variously described above in connectionwith method 1300.

The DAC unit 1415 may be configured to convert the resultant digital Isignal and the resultant digital Q signal to analog form in order torespectively obtain an analog I signal and an analog Q signal.

The I/Q modulator 1420 may be configured to perform I/Q modulation onthe analog I and Q signals to produce a modulated analog signal. The 2×2matrix is configured to at least partially pre-compensate for I/Qimpairments of the transmitter at frequency f. A first of the constants,corresponding a first diagonal element of the 2×2 matrix, may becomputed based on a measurement of the I/Q impairments at frequency fand a measurement of the I/Q impairments at frequency −f. A second ofthe constants, corresponding to a first non-diagonal element of the 2×2matrix, may be computed based on the measurement at frequency f and themeasurement at frequency −f.

The Meaning of “Impairments at Frequency f”

The present disclosure repeatedly uses the term “the I/Q impairments atfrequency f”. Whether that term is applied to the transmitter, thereceiver, or the series combination comprising the transmitter, thetransmission path and the receiver, it includes within its scope ofmeaning the I/Q impairments that result from stimulating the system inquestion with a complex exponential tone exp(j2πft)=cos(2πft)+jsin(2πft) at frequency f, as shown in FIG. 15. The real and imaginaryoutputs of the system may be represented as:

I′(t)=g _(I)(f)cos(2πft+φ _(I)(f))

Q′(t)=g _(Q)(f)sin(2πft+φ _(Q)(f)).

The I/Q impairments at frequency f may include the gain imbalance g(f)and the phase skew φ(f) given by:

g(f)=g _(Q)(f)/g _(I)(f)

φ(f)=φ_(Q)(f)−φ_(I)(f).

Here we adopt the convention of using the I channel as the reference forboth the gain imbalance and the phase skew. However, the inventiveprinciples described herein apply equally to any other referencingconvention. For example, one might just as well use the oppositeconvention (i.e., the choice of the Q channel as reference for both thegain imbalance and the phase skew), or a convention where the gainimbalance is referenced to one channel and the phase skew to the other.

Because we are interested in compensating for imbalances of gain anddifferences in phase between the two channels, we can model the gainimbalance and the phase skew as appearing all on the I channel or all onthe Q channel. For example, FIG. 16 illustrates the latter choice. Thus,the Q channel output has the form:

Q′(t)=g(f)sin(2πft+φ(f)).

The Physical Consequence of I/Q Impairments

The consequence of I/Q impairments at frequency f is the appearance ofunwanted signal energy at the frequency −f. To see this, we analyze thecomplex output signal as follows:

$\begin{matrix}{{{I^{\prime}(t)} + {j\; {Q^{\prime}(t)}}} = {{\cos \left( {\omega \; t} \right)} + {j\; {g(\omega)}{\sin \left( {{\omega \; t} + {\phi (\omega)}} \right)}}}} \\{= {{\left( {1/2} \right)\left\{ {{\exp \left( {j\; \omega \; t} \right)} + {\exp \left( {{- j}\; \omega \; t} \right)}} \right\}} + {\left( {1/2} \right){g(\omega)}}}} \\{\left\{ {{\exp \left( {{j\; \omega \; t} + {\phi (\omega)}} \right)} - {\exp \left( {{{- j}\; \omega \; t} - {\phi (\omega)}} \right)}} \right\}} \\{= {{\left( {1/2} \right)\left\{ {1 + {{g(\omega)}{\exp \left\lbrack {\phi (\omega)} \right\rbrack}}} \right\} {\exp \left( {j\; \omega \; t} \right)}} + \left( {1/2} \right)}} \\{{\left\{ {1 - {{g(\omega)}{\exp \left\lbrack {- {\phi (\omega)}} \right\rbrack}}} \right\} {\exp \left( {{- j}\; \omega \; t} \right)}}} \\{= {{{A_{TONE}(\omega)}{\exp \left( {j\; \omega \; t} \right)}} + {{A_{IMAGE}(\omega)}{{\exp \left( {{- {j\omega}}\; t} \right)}.}}}}\end{matrix}$

(We have switched from f to ω=2πf merely for the sake of notationalbrevity.) Thus, in response to the stimulus signal exp(jωt), the systemproduces a complex exponential tone at frequency ω with complexamplitude A_(TONE)(ω) and a complex exponential tone at frequency −ωwith complex amplitude A_(IMAGE)(ω).

A _(TONE)=(½){1+g(ω)exp[φ(ω)]}

A _(IMAGE)=(½){1−g(ω)exp[−φ(ω)]}.

The complex exponential tone at frequency ω is often referred to simplyas “the tone”, while the complex exponential tone at frequency −ω isoften referred to as “the image”. As expected, A_(TONE)(ω)→1 andA_(IMAGE)(ω)→0 as g(ω)→1 and φ(ω)→0. It is desirable to have g(ω) asclose to one as possible and φ(ω) as close to zero as possible. (Linearscale is being assumed here for the gain imbalance. Gain imbalance mayalso be expressed in a logarithmic scale, e.g., in units of dB, in whichcase 0 dB represents the case of no gain imbalance.)

From the above discussion, one can readily see that the seriescombination of two systems, the first having gain imbalance g₁(ω) andphase skew φ₁(ω), and the second having gain imbalance g₂(ω) and phaseskew φ₂(ω), does not give a net gain imbalance of g(ω)=g₁(ω)g₂(ω) and anet phase skew of φ(ω)=φ₁(ω)+φ₂(ω). (This is because the second systemis not being stimulated by a pure complex exponential exp{jωt}.) Thetrue relations are more complicated.

Image Rejection

Image rejection is a measure of the relative magnitudes of complexamplitudes A_(TONE)(ω) and A_(IMAGE)(ω). For example, according to oneconventional definition:

Image_Rejection=20*log(|A _(IMAGE) |/|A _(TONE)|).

Because |A_(IMAGE)| is typically smaller than |A_(TONE)|, the imagerejection is typically negative. The more negative the image rejection,the better.

Post-Compensation Vs. Pre-Compensation

The notion of post-compensation involves coupling a compensation blockto the output of a system that exhibits I/Q impairments. Thecompensation block is configured so that the series combination of thesystem followed by the compensation block behaves (or, approximates thebehavior of) an ideal model having unity gain imbalance and zero phaseskew. When the system is stimulated by a complex exponential tone atfrequency ω, it will generate a distorted complex signal that can bemodeled as

cos(ωt)+jg(ω)sin(ωt+φ(ω)),

where g(ω) and φ(ω) are the I/Q impairments of the system at frequencyω. The compensation block operates on the distorted complex signal togenerate a corrected output signal equal to the original complexexponential tone at frequency ω. Thus, the compensation block is said to“compensate” or “post-compensate” for the I/Q impairments of the systemat frequency ω. Wideband post-compensation of I/Q impairments meanspost-compensating for I/Q impairments at each frequency ω in a frequencyrange or frequency band.

The notion of pre-compensation involves placing a compensation blockbefore the system, i.e., the output of the compensation block couples tothe input of the system. The compensation block is configured so thatthe series combination of the compensation block followed by the systembehaves (or, approximates the behavior of) an ideal model having unitygain imbalance and zero phase skew. In response to a complex exponentialtone at frequency ω, the compensation block will produce a pre-distortedcomplex signal. The system receives the pre-distorted complex signal andfurther distorts that signal (by introducing I/Q impairments), therebyproducing a complex output signal. The compensation block generates thepre-distorted complex signal so that the complex output signal from thesystem is equal to the original complex exponential tone at frequency ω.Thus, the compensation block is said to “compensate” or “pre-compensate”for the I/Q impairments of the system at frequency ω. Widebandpre-compensation of I/Q impairments means pre-compensating for I/Qimpairments at each frequency ω in a frequency range or frequency band.

Performing Traditional Impairment Compensation at a Single Frequency

If one is interested in post-compensating for I/Q impairments at aparticular frequency ω₀, one might use the block diagram of FIG. 17 withreal constants α and β. By appropriate choice of the constants, thedisturbed complex input signal

cos(ω₀ t)+jg(ω₀)sin(ω₀ t+φ(ω₀))

will be mapped to the corrected output signal cos(ω₀t)+j sin(ω₀t), asdesired. The appropriate values are:

α=−tan(φ(ω₀))

β=1/{g(ω₀)cos(φ(ω₀))}.

This method of compensation is referred to herein as “traditionalsingle-point compensation”.

Due to the continuity of the gain imbalance g and phase skew φ withrespect to frequency ω, the real constants α and β will achieve partialcompensation for I/Q impairments at frequencies in a neighborhood of ω₀,with the quality of compensation degrading with increasing distance fromω₀. However, because g(ω₀) is typically different from g(−ω₀), and φ(ω₀)is typically different from φ(−ω₀), the appropriate pair of values (α,β)for compensating I/Q impairments at frequency ω₀ is typically differentfrom the appropriate pair of values for compensating I/Q impairments atfrequency −ω₀. Thus, unfortunately, one typically cannot find a singlepair of values that will work for both ω₀ and −ω₀.

While the above-derived values of α and β work perfectly forpost-compensation of I//Q impairments at the single frequency ω₀, theymay also be used for pre-compensation of I/Q impairments at the singlefrequency ω₀ with generally less than perfect results. (Various ones ofthe methods described herein may employ such pre-compensation, eventhough it gives less than perfect results, in part because it does notrequire knowledge of the I/Q impairments at frequency −ω₀.) To achieveperfect pre-compensation of I/Q impairments at a single frequency, seethe section “Calculating True Single Point Vector CalibrationConstants”.

Wideband I/Q Impairment Equalization

FIG. 18 depicts a basic model of a system H that will be used repeatedlythroughout this patent disclosure, e.g., to represent the equalizationfiltering performed by the receiver and the equalization filteringperformed by the transmitter. (Equalization is used here as a synonymfor I/Q impairment compensation.)

In the case where system H represents the receiver's equalizationfiltering, the complex input signal I(t)+jQ(t) represents a distortedsignal supplied by a preceding system G, as illustrated in FIG. 19. Thesystem G generates the distorted signal

I(t)+jQ(t)=cos(2πft)+jg(f)sin(2πft+φ(f))

in response to being stimulated by the complex exponential signali(t)+jq(t)=exp(j2πft) at frequency f. The gain imbalance g(f) and thephase skew φ(f) are the I/Q impairments of the system G at frequency f.System G may represent the baseband equivalent of the receiver's frontend, i.e., the portions of the receiver from its RF input to the outputof the I/Q digitization unit. Alternatively, in the situation where thereceiver is expected to compensate for the transmitter's I/Q impairmentsas well as for its own, system G may represent the path from the inputof the transmitter's I/Q DAC unit to the output the receiver I/Qdigitization unit.

The system H operates on the distorted input to produce a correctedoutput signal I′(t)+jQ′(t)=exp(j2πft), for all f in a desired frequencyband. Note, however, that the set B given by

{exp(j2πft): f in given frequency range}

forms a basis for the space of functions {x(t)} that are band limited tothe given frequency range. Because the series combination of G followedby H is an identity map on each function of the basis set B, bylinearity it will be an identity map on all band-limited functions x(t).

The equalization system H may be realized by the receiver's digitalcircuit 220 as variously described above.

In the case where system H represents the transmitter's equalizationfiltering, we interpret H as receiving a basis function

I(t)+Q(t)=cos(2πft)+jg(f)sin(2πft+φ(f)),

and, in response to that basis function, generating a pre-compensatedcomplex signal

I′(t)+jQ′(t)=exp(j2πft),

as shown in FIG. 20A. Note that the set X given by

{cos(2πft)+jg(f)sin(2πft+φ(f)): f in given frequency range}

also forms a basis for the space of functions {x(t)} that are bandlimited to the given frequency range.

The pre-compensated signal that gets distorted by the following systemG. The system G generates distorted signal

i(t)+jq(t)=cos(2πft)+jg(f)sin(2πft+φ(f)),

where g(f) and φ(f) represent the gain imbalance and phase skew of thesystem G at frequency f. Because the series combination of H followed byG is an identity map on each function of the basis set X, it will be anidentity map on all band-limited functions x(t). Thus, when stimulatedby a complex exponential tone exp(j2πft) at any frequency f in thefrequency band, the series combination will produce that same complexexponential tone at its final output, as shown in FIG. 20B.

The system G may represent the baseband equivalent of the transmitter'sRF front end, i.e., the portions of the transmitter from the input ofthe transmitter's DAC unit to the RF output. Alternatively, insituations where the transmitter is expected to compensate for thereceiver's I/Q impairments as well as for its own, the system G mayrepresent the path from the input of the transmitter's DAC unit to theoutput of the receiver's digitization unit. The system H may be realizedby the digital circuit 510 as variously described above.

Complex exponentials are used throughout the present analysis since anyband-limited signal can be represented as the superposition of anensemble of complex exponentials via Fourier analysis. I/Q impairmentsmay include an imbalance of the gain when comparing the in-phase (I)channel with the quadrature-phase (Q) channel, and a skew in the phasethat occurs due to imperfect quadrature mixing. (The phase skew perturbsthe ideal 90-degree phase relationship between the I and Q channels).While phase skew is typically modeled as an imperfection in thequadrature mixing, it can also be modeled as the phase skew between theI(t) and Q(t) signals. In both cases discussed above, the input to thedistortion model G is a complex exponential signal. Since I/Qimpairments are relative, we may assume that the I/Q impairments appearentirely at the Q(t) output while the I(t) output is ideal. While otherassumptions could be made, this assumption will simplify the followingmathematical derivations.

The equalization system H may be modeled by a 2×2 frequency responsematrix H(f)=(H_(ij)(f)), or equivalently, by a 2×2 matrixh(t)=(h_(ij)(t)) of real-valued impulse responses. However, under theabove-identified assumption on how the impairments are expressed at theoutput of the distortion model G, the matrix H may be simplified to thestructure shown in FIG. 21, i.e., H₁₁(f)=1 and H₁₂(f)=0. For notationalefficiency, we define U(f)=H₂₁(f) and V(f)=H₂₂(f). Thus,

I′(t)=I(t)

Q′(t)=u(t)*I(t)+v(t)*Q(t),

where u(t) and v(t) are the impulse responses corresponding to U(f) andV(f) respectively.

Any real-valued filter must necessarily have a symmetric magnituderesponse and an anti-symmetric phase response. In other words, x(t)being real implies

|X(f)|=|X(−f)|

Phase{X(−f)}=−Phase{X(f)}

for all f, where X(f) is the Fourier transform of x(t). As a result, thefrequency response V(f) cannot apply independent impairment correctionsat frequencies f and −f. In the typical situation g(f) and g(−f) aredifferent, and φ(f) and φ(−f) are different. Thus, the filter V actingby itself (i.e., with U identically equal to zero) is not sufficient toprovide correction at f and −f. If the goal was to only correct for thewideband I/Q impairments over positive frequencies only or over negativefrequencies only, the filter V would suffice. (Note: V acting by itselfcan correct for +f and −f impairments as long as the impairments areconstrained to g(f)=g(−f) and φ(f)=−φ(−f), as demonstrated in the“Adding Constraints” section.) However, since it is desirable to correctfor both sides of the spectrum, the second filter U(f) is introduced.Applying another filter from in-phase component and summing it into thequadrature-phase channel provides the needed degrees of freedom tocontrol both sides of the complex spectrum. This is due to the fact thatthe in-phase component I(t)=cos(2πft) is that same for frequencies f and−f, while the quadrature-phase component changes its phase by 180degrees when changing from f to −f.

In order to solve for U(f) and V(f), their respective output signalsneed to be known. In order to simplify the mathematical derivation, bothU(f) and V(f) are broken into their even and odd parts as shown in FIG.22. Thus, A(f) and B(f) are the even and odd parts of U(f), while C(f)and D(f) are the even and odd parts of V(f).

Since any real-valued filter must necessarily have a symmetric magnituderesponse, we can reduce complexity by solving only for thepositive-frequency portion of each spectrum A, B, C and D. However, toachieve impairment compensation for negative frequencies as well aspositive frequencies, one cannot simply ignore the inputs I(t)+jQ(t)corresponding to negative frequencies. Rather, we account for suchinputs by expressing them as equivalent positive-frequency inputs,relying on the odd symmetry of the sine function and the even symmetryof the cosine function:

$\begin{matrix}{{{I(t)} + {j\; {Q(t)}}} = {{\cos \left( {{- 2}\pi \; f\; t} \right)} + {{g\left( {- f} \right)}{\sin \left( {{{- 2}\pi \; f\; t} + {\phi \left( {- f} \right)}} \right)}}}} \\{= {{\cos \left( {2\pi \; f\; t} \right)} - {{g\left( {- f} \right)}{{\sin \left( {{2\pi \; f\; t} - {\phi \left( {- f} \right)}} \right)}.}}}}\end{matrix}$

Thus, we shall develop two equations for the positive-frequency portionsof A, B, C and D, the first based on the input

I ₁(t)+jQ ₁(t)=cos(2πft)+g(f)sin(2πft+φ(f)),

and the second based on the input

I ₂(t)+jQ ₂(t)=cos(2πft)−g(−f)sin(2πft−φ(−f)),

with f>0 for both equations.

If a filter is constrained to have symmetric impulse response, then thefilter will exhibit a symmetric magnitude response and a zero phaseresponse. Such is the case for filters a(t) and c(t). If, however, afilter's impulse response is anti-symmetric, then it will exhibit asymmetric magnitude response but a phase response that equals

−(π/2)sgn(f).

Thus, an anti-symmetric impulse response is equivalent to an evenimpulse response followed by a Hilbert transform. Such is the case forthe filters b(t) and d(t). As a result, filter b(t) can be expressed asan even impulse response e_(B)(t) followed by a Hilbert transform (HT),as shown in FIG. 23. Similarly, filter d(t) can be expressed as an evenimpulse response e_(D)(t) followed by a Hilbert transform (HT). E_(B)(f)and E_(D)(f) are the frequency responses corresponding to e_(B)(t) ande_(D)(t), respectively. Now the exact outputs of the original filters A,B, C and D can easily be determined. FIG. 24A shows the outputs of thefour filters A, B, C and D in response to signal I₁(t)+jQ₁(t). FIG. 24Bshows the outputs of the four filter in response to the signalI₂(t)+jQ₂(t).

Each of FIGS. 24A and 24B can be directly translated into acorresponding linear equation in A(f), E_(B)(f), C(f) and E_(D)(f) forpositive f (or non-negative f). We use the following notation:

g ₁(f)=g(f) for f>0

g ₂(f)=g(−f) for f>0

φ(f)=φ(f) for f>0

φ₂(f)=φ(−f) for f>0.

FIGS. 24A and 24B give rise, respectively, to equations (1.1) and (1.2),which are shown in FIG. 25. FIGS. 26A and 26B present the correspondingphasor diagrams. (Recall that cos(2πft) maps to 1, and sin(2πft) maps to−j in the phasor diagram.)

The horizontal projections of the vectors in FIG. 26A give equation(1.3) below; the vertical projections give equation (1.4). Similarly,the horizontal projections of the vectors in FIG. 26B give equation(1.5), while the vertical projections give equation (1.6):

0=A(f)+C(f)g ₁(f)sin(φ₁(f))−E _(D)(f)g ₁(f)cos(φ₁(f))  (1.3)

−1=−E _(B)(f)−C(f)g ₁(f)cos(φ₁(f))−E _(D)(f)g ₁(f)sin(φ₁(f))  (1.4)

0=A(f)+C(f)g ₂(f)sin(φ₂(f))+E _(D)(f)g ₂(f)cos(φ₂(f))  (1.5)

1=−E _(B)(f)+C(f)g ₂(f)cos(φ₂(f))−E _(D)(f)g ₂(f)sin(φ₂(f))  (1.6)

This system of equations is a 4×4 linear system in unknown vector(A,E_(B),C,E_(D)):

$\begin{matrix}{{\begin{bmatrix}0 \\{- 1} \\0 \\1\end{bmatrix} = {\lbrack P\rbrack \begin{bmatrix}{A(f)} \\{E_{B}(f)} \\{C(f)} \\{E_{D}(f)}\end{bmatrix}}},{where}} & (1.7) \\{P = \begin{bmatrix}1 & 0 & w & {- x} \\0 & {- 1} & {- x} & {- w} \\1 & 0 & y & z \\0 & {- 1} & z & {- y}\end{bmatrix}} & (1.8)\end{matrix}$

and

w=g ₁(f)sin(φ₁(f))  (1.9)

x=g ₁(f)cos(φ₁(f))  (1.10)

y=g ₂(f)sin(φ₂(f))  (1.11)

z=g ₂(f)cos(φ₂(f)).  (1.12)

The determinate of matrix P is given by:

Det(P)=w ² +x ² +y ² +z ²−2wy+2xz.  (1.13)

Det(P)=g ₁ ²(f)+g ₂ ²(f)+2g ₁(f)g ₂(f)cos(φ₁(f)+φ₂(f))  (1.14)

As long as

g ₁ ²(f)+g ₂ ²(f)+2g ₁(f)g ₂(f)cos(φ₁(f)+φ₂(f))≠0,  (1.15)

there exists a unique solution vector (A(f),E_(B)(f),C(f),E_(D)(f)). Asan example, the equations cannot be solved when both the phase skew φ(f)and gain imbalance g(f) are completely odd. However, it does not makesense for the gain imbalance g(f) to be completely odd since the gainimbalance is typically close to one for all f, or at least bounded belowby a positive constant.

Using Cramer's Rule, we find that

A(f)=−2(wz+xy)/Det(P).  (1.16)

E _(B)(f)=(−w ² −x ² +y ² +z ²)/Det(P)  (1.17)

C(f)=2(x+z)/Det(P)  (1.18)

E _(D)(f)=2(w−y)/Det(P).  (1.19)

Substituting equation (1.9) through (1.14) into equations (1.16) through(1.19) yields equations (1.20) through (1.23), shown in FIG. 27.

Adding Constraints

In many cases, the gain imbalance and the phase skew approximate commonconstraints. This section simplifies equations (1.20) through (1.23) forsome typical real world conditions. For the most perfect compensation,the equations (1.20-1.23) may be used. But if the compensationperformance can be relaxed, then adding some constraints can decreasethe computational requirements.

Case 1: Odd Phase Skew

In the case of odd phase skew, i.e., φ(f)=φ₁(f)=−φ₂(f) for f>0,equations (1.20) through (1.23) specialize to:

A(f)=0  (1.24)

E _(B)(f)={g ₂(f)−g ₁(f)}/{g ₁(f)+g ₂(f)}  (1.25)

C(f)=2 cos(φ(f))/{g ₁(f)+g ₂(f)}  (1.26)

E _(D)(f)=2 sin(φ(f))/{g ₁(f)+g ₂(f)}.  (1.27)

Case 2: Even Gain Imbalance

In the case of even gain imbalance, i.e., g(f)=g₁(f)=g₂(f) for f>0,equations (1.20) through (1.23) specialize to:

$\begin{matrix}{{A(f)} = \frac{- {\sin \left( {{\phi_{1}(f)} + {\phi_{2}(f)}} \right)}}{1 + {\cos \left( {{\phi_{1}(f)} + {\phi_{2}(f)}} \right)}}} & (1.28) \\{{E_{B}(f)} = 0} & (1.29) \\{{C(f)} = \frac{{\cos \left( {\phi_{1}(f)} \right)} + {\cos \left( {\phi_{2}(f)} \right)}}{{g(f)}\left\{ {1 + {\cos \left( {{\phi_{1}(f)} + {\phi_{2}(f)}} \right)}} \right\}}} & (1.30) \\{{E_{D}(f)} = {\frac{{\sin \left( {\phi_{1}(f)} \right)} - {\sin \left( {\phi_{2}(f)} \right)}}{{g(f)}\left\{ {1 + {\cos \left( {{\phi_{1}(f)} + {\phi_{2}(f)}} \right)}} \right\}}.}} & (1.31)\end{matrix}$

Case 3: Odd Phase Skew and Even Gain Imbalance

In the case of odd phase skew and even gain imbalance, equations (1.20)through (1.23) specialize to:

A(f)=0  (1.32)

E _(B)(f)=0  (1.33)

C(f)=cos(φ(f))/g(f)  (1.34)

E _(D)(f)=sin(φ(f))/g(f).  (1.35)

Case 4: Zero Phase Skew and Arbitrary Gain Imbalance

In the case of zero phase skew and arbitrary gain imbalance, equations(1.20) through (1.23) specialize to:

A(f)=0  (1.36)

E _(B)(f)={g ₂(f)−g ₁(f)}/{g ₂(f)+g ₁(f)}  (1.37)

C(f)=2/{g ₂(f)+g ₁(f)}  (1.38)

E _(D)(f)=0.  (1.39)

Case 5: Arbitrary Phase Skew and Unity Gain Imbalance

In the case of arbitrary phase skew and unity gain imbalance, equations(1.20) through (1.23) specialize to:

$\begin{matrix}\begin{matrix}{{A(f)} = \frac{- {\sin \left( {{\phi_{1}(f)} + {\phi_{2}(f)}} \right)}}{1 + {\cos \left( {{\phi_{1}(f)} + {\phi_{2}(f)}} \right)}}} \\{= {- {\tan \left( \frac{{\phi_{1}(f)} + {\phi_{2}(f)}}{2} \right)}}}\end{matrix} & (1.40) \\{{E_{B}(f)} = 0} & (1.41) \\{{C(f)} = \frac{{\cos \left( {\phi_{1}(f)} \right)} + {\cos \left( {\phi_{2}(f)} \right)}}{1 + {\cos \left( {{\phi_{1}(f)} + {\phi_{2}(f)}} \right)}}} & (1.42) \\{{E_{D}(f)} = {\frac{{\sin \left( {\phi_{1}(f)} \right)} - {\sin \left( {\phi_{2}(f)} \right)}}{1 + {\cos \left( {{\phi_{1}(f)} + {\phi_{2}(f)}} \right)}}.}} & (1.43)\end{matrix}$

Case 6: Constant Gain Imbalance and Phase Skew

In the case where the gain imbalance and phase skew functions areconstant functions, i.e., g(f)=g and φ(f)=φ over all f, equations (1.20)through (1.23) specialize to:

$\begin{matrix}{{A(f)} = {\frac{- {\sin \left( {2\phi} \right)}}{1 + {\cos \left( {2\phi} \right)}} = {- {\tan (\phi)}}}} & (1.44) \\{{E_{B}(f)} = 0} & (1.45) \\{{C(f)} = {\frac{2\; {\cos (\phi)}}{g\left\{ {1 + {\cos \left( {2\phi} \right)}} \right\}} = \frac{1}{g\; {\cos (\phi)}}}} & (1.46) \\{{E_{D}(f)} = 0.} & (1.47)\end{matrix}$

Filter Design

In one embodiment, symmetric linear-phase FIR filters â(n) and ĉ(n) aredesigned based respectively on the magnitude responses |A(f)| and|C(f)|, while antisymmetric linear-phase FIR filters {circumflex over(b)}(n) and {circumflex over (d)}(n) are designed based respectively onthe magnitude responses |B(f)| and |D(f)|. Note that |B(f)|=|E_(B)(f)|and |D(f)|=|E_(D)(f)| for all f. The Remez algorithm may be used todesign these filters. The equalization system of FIG. 22 may then beimplemented using the filters â(n), {circumflex over (b)}(n), ĉ(n) and{circumflex over (d)}(n). By creating four filters, each with eithersymmetric or anti-symmetric filter taps, and summing the filters asshown in FIG. 22, we can effectively match the two arbitrary frequencyresponses U(f) and V(f). (Note: Depending on the filter design tool, thesummation might actually be a subtraction. The definition of HilbertTransform used by many filter design tools to create anti-symmetricfilters differ from the definition we have used by a negation. Thefilter design tools often use +(π/2)sgn(f) for the phase.)

In another embodiment, symmetric linear-phase FIR filters â(n),ê_(B)(n), ĉ(n) and ê_(D)(n) are designed based respectively on themagnitude responses |A(f)|, |E_(B)(f)|, |C(f)| and |E_(D)(f)|. Again,the Remez algorithm may be used to design these filters. Theequalization system of FIG. 23 may then be realized using the filtersâ(n), ê_(B)(n), ĉ(n) and ê(n).

In yet another embodiment, filters û(n) and {circumflex over (v)}(n) maybe designed based on the frequency responses U(f) and V(f). AnL_(p)-norm design method may be used to design these filters based onthe magnitude and phase responses of U(f) and the magnitude and phaseresponses of V(f). The equalization system of FIG. 21 may then beimplemented using the filters û(n) and {circumflex over (v)}(n).

Corruption I/Q Impairments

As described above, FIG. 15 illustrates a system that introduces I/Qimpairments, i.e., gain imbalance g(f) and phase skew φ(f), into areceived complex exponential signal exp(j2πft). In general, a 2×2frequency response matrix H characterizing the system may be derivedfrom the impairment functions g(f) and φ(f). To simplify thisderivation, we model the gain imbalance g(f) and phase skew φ(f) of thesystem as appearing entirely on the Q channel output, as shown in FIG.28. This model makes it convenient to use the special form of the matrixshown in FIG. 29, where U(f) and V(f) are frequency responsescorresponding to real filters u(t) and v(t). Response U(f) may berepresented as the sum of its even part A(f) and odd part B(f), as shownin FIG. 30. Similarly, V(f) may be represented as the sum of its evenpart C(f) and odd part D(f). The filter with odd spectrum B(f) may berepresented by a subsystem with even spectrum E_(B)(f) followed by aHilbert Transform HT, as shown in FIG. 31. (See the above “NoteRegarding Filters with Odd Frequency Response”.) Similarly, the filterwith odd spectrum D(f) may be represented by a subsystem with evenspectrum E_(D)(f) followed by a Hilbert Transform HT. Note that themagnitude responses of B and E_(B) are identical, |B(f)|=|E_(B)(f)|, asare the magnitude responses of D(f) and E_(D)(f).

We will develop equations for the positive-frequency portions of A(f),E_(B)(f), C(f) and E_(D)(f) since the negative-frequency portions aredetermined by the respective positive frequency portions. One equationwill come from stimulating the system with a positive frequency inputI₁(t)+jQ₁(t)=exp(j2πft) for f>0, as shown in FIG. 32A. Another equationwill come from stimulating the system with a negative frequency inputthat is expressed in terms of an equivalent positive frequency input:

$\begin{matrix}{{{I_{2}(t)} + {j\; {Q_{2}(t)}}} = {\exp \left( {{- {j2\pi}}\; {ft}} \right)}} \\{= {{\cos \left( {{- 2}\pi \; {ft}} \right)} + {j\; {\sin \left( {{- 2}\pi \; {ft}} \right)}}}} \\{= {{\cos \left( {2\pi \; {ft}} \right)} - {j\; {\sin \left( {2\pi \; {ft}} \right)}}}}\end{matrix}$

for f>0, as shown in FIG. 32B.

FIG. 33 shows the two equations. Equation (1.48) is based on FIG. 32A.Equation (1.49) is based on FIG. 32B.

FIGS. 34A and 34B show the corresponding phasor diagrams, relying on thefollowing notation:

g ₁(f)=g(f) for f>0

g ₂(f)=g(−f) for f>0

φ₁(f)=φ(f) for f>0

φ₂(f)=φ(−f) for f>0.

The phasor diagrams give the following equations:

g ₁(f)sin(φ₁(f))=A(f)−E _(D)(f)  (1.50)

g ₁(f)cos(φ₁(f))=E _(B)(f)+C(f)  (1.51)

g ₂(f)sin(φ₂(f))=A(f)+E _(D)(f)  (1.52)

g ₂(f)cos(φ₂(f))=C(f)−E _(B)(f)  (1.53)

These equations comprise a 4×4 matrix equation (1.54) in the unknownsA(f), E_(B)(f), C(f) and E_(D)(f), as shown in FIG. 35. By inverting the4×4 coefficient matrix, we obtain the solution. See the matrix equation(1.55) in FIG. 36. It follows that:

A(f)=(½){g ₁(f)sin(φ₁(f))+g ₂(f)sin(φ₂(f))}  (1.56)

E _(B)(f)=(½){g ₁(f)cos(φ₁(f))−g ₂(f)cos(φ₂(f))}  (1.57)

C(f)=(½){g ₁(f)cos(φ₁(f))+g ₂(f)cos(φ₂(f))}  (1.58)

E _(D)(f)=(½){−g ₁(f)sin(φ₀(f))+g ₂(f)sin(φ₂(f))}.  (1.59)

Since A, E_(B), C and E_(D) are even functions of frequency f. Thus,their negative-frequency portions are specified by the even symmetry.Furthermore, the odd frequency responses B(f) and D(f) are given by:

B(f)=−jE _(B)(f)sgn(f) and

D(f)=−jE _(D)(f)sgn(f).

Special Case: Even Gain Imbalance and Odd Phase Skew

In the case where the gain imbalance is an even function and the phaseskew is an odd function, i.e., g(f)=g(−f) and φ(f)=−φ(−f), thenequations (1.56) through (1.59) specialize to:

A(f)=0

E _(B)(f)=0

C(f)=g(f)cos(φ(f))

E _(D)(f)=−g(f)sin(φ(f)).

Special Case: Even Gain Imbalance and Even Phase Skew

In the case where the gain imbalance and phase skew are even functions,i.e., g(f)=g(−f) and φ(f)=φ(−f), then equations (1.56) through (1.59)specialize to:

A(f)=g(f)sin(φ(f))

E _(B)(f)=0

C(f)=g(f)cos(φ(f))

E _(D)(f)=0.

Special Case: Constant Gain Imbalance and Phase Skew

In the case where the gain imbalance and phase skew are constantfunctions, i.e., g(f)=g and φ(f)=φ, then equations (1.56) through (1.59)specialize to:

A(f)=g sin(φ)  (1.60)

E _(B)(f)=0  (1.61)

C(f)=g cos(φ)  (1.62)

E _(D)(f)=0.  (1.63)

Calculating the Mapping Between the Rx and Tx

In some embodiments, a transmitter pre-distorts digital I/Q signals inorder to compensate for its own I/Q impairments as variously describedabove. To accomplish this compensation, one must have an estimate of thetransmitter's I/Q impairments. The quality of the compensation will belimited by the quality of the estimate (the extent to which it matchesthe truth). While a high-quality estimate is desirable, it is difficultto measure the transmitter's I/Q impairments directly. Rather, themeasurements are obtained indirectly, e.g., using a receiver as shown inFIG. 37.

FIG. 37 shows a transmitter 3700 that couples to a receiver 3725 via achannel (e.g., a cable 3720 or a wireless channel). The transmitter mayinclude a digital compensation unit 3702, a DAC unit 3705, an I/Qmodulator 3710 and a front end 3715. The compensation unit 3702 mayperform pre-compensation (pre-distortion) on the digital signalI(n)+jQ(n) to obtain a pre-compensated signal I′(n)+jQ′(n), e.g., asvariously described above. DAC unit 3705 may convert the pre-compensatedsignal into an analog signal s(t)=I′(t)+jQ′(t). The analog signal s(t)may be upconverted to RF using the I/Q modulator 3710. The upconvertedsignal may be conditioned by the TX front end 3715 to obtain a transmitsignal. The transmit signal may be delivered to the receiver by thecable 3720.

The receiver 3725 may include a front end 3730, an I/Q demodulator 3735and a digitization unit 3740. The front end 3830 may receive thetransmitted signal from the cable 3720 and operate on the receivedsignal to produce a conditioned signal. The conditioned signal may bedown-converted by the I/Q demodulator to produce a complexdown-converted signal. The complex down-converted signal may be sampledby the digitization unit 3740 to obtain a sampled complex signal. Thesampled complex signal may be used to make I/Q impairment measurements.In some embodiments, the receiver is a spectrum analyzer, e.g., a vectorsignal analyzer.

It is important to understand how measurements of I/Q impairments takenat the receiver 3725 relate to the transmitter's I/Q impairments. Theyare not the same. This is because the I/Q impairments at thetransmitter, e.g., at the output of the I/Q modulator, are obscured(distorted) by the signal path including the TX front end 3715, thecable 3720 and the receiver front end 3730. The signal path may becharacterized by a frequency response H(f)=m(f)exp(jθ(f)), where H(f) isa complex number. The amplitude m(f) is referred to herein as the“scaling” of the signal path at frequency f. The phase θ(f) is referredto herein as the “rotation” of the signal path at frequency f.

The problem of estimating the transmitter's I/Q impairments fromreceiver-based measurements is not trivial. Its solution is disclosed inthis patent disclosure. (See the iterative methods disclosed below.)Part of the solution includes obtaining an initial estimate for thesignal path response function H(f). This section will focus on obtainingthat initial estimate in the form of H(0), i.e., the frequency responseof the signal path at DC (zero frequency). The amplitude m(0) of H(0) isreferred to as the “DC scaling” of the signal path. The phase θ(0) ofH(0) is referred to as the “DC rotation” of the signal path.

One way to estimate the transmitter's I/Q impairments involvesperforming an iterative procedure using a spectrum analyzer. (A spectrumanalyzer is a device configured to measure the magnitude of an inputsignal versus frequency within the frequency range of the instrument.)The spectrum analyzer measures the I/Q impairments of its demodulatedsignal, and then compensation is applied at the transmitter based onthat measurement. The measurement may only crudely approximate thetransmitter's I/Q impairments, but it may be good enough to achieve atleast partial compensation. The spectrum analyzer then makes a secondmeasurement of the I/Q impairments of its demodulated signal. Thissecond measurement may be used to make an adjustment to the compensationbeing applied at the transmitter, and so on. The sequence ofmeasurements may converge, i.e., the measured gain imbalance mayconverge to one and the measured phase skew may converge to zero,indicating that appropriate compensation has been achieved at thetransmitter. Because the spectrum analyzer does not capture phaseinformation, multiple iterations may be required to achieve convergence.

In some embodiments, the transmitter's I/Q impairments may be determinedusing a measurement device (such as a vector signal analyzer) that canmake phase measurements and can lock the phase of the measurements tothe transmitter. In this case, the transmitter's I/Q impairments may bedetermined with two measurements or less at the measurement device.

The method described below makes two measurements, but requires that thetransmitter's I/Q modulator and the receiver's I/Q demodulator be lockedtogether in frequency (via phase locked loops with common reference).Unlike other methods, this method is resistant to synchronous spurs(i.e., spurs such as the LO leakage that are phase locked to the LO ofthe transmitter). While this technique can be used at any frequency, themain application is to determine the DC scaling m(0) and the DC rotationθ(0) of the signal path in order to calibrate out the transmitter's LOleakage impairment.

In FIG. 38, vector A=A₁+jA_(Q) represents the transmitter's LO leakagewhen the transmitter is stimulated with a constant zero signal:I′(n)=Q′(n)=0. (The term “vector” is used here as a synonym for “complexnumber”.) The amplitude and phase of vector A represent the amplitudeand phase of the LO leakage. As this LO leakage signal moves from thetransmitter's I/Q modulator to the receiver's I/Q demodulator, it getsscaled by m(0) and rotated by θ(0) so that vector A is transformed intovector A′ at the receiver. See FIG. 39. The vector A′ is measured, e.g.,by averaging the sampled complex signal captured from the output of theI/Q demodulator.

Then we stimulate the transmitter with a known non-zero vector B:

I′(n)=B _(I)

Q′(n)=B _(Q).

(Vector B need not be real as shown in FIG. 38. However, it does need tobe non-zero.) This intentionally-applied LO leakage B is superimposed ontop of the transmitter's intrinsic LO leakage A so that thetransmitter's total leakage is vector C. (The choice of B, primarily itsmagnitude, can affect the accuracy of the measurement. The optimum sizewill depend on the specific hardware. If it is too small, noise willinfluence the measurement more; if it is too big, the hardware could beput into a non-linear region of operation.) This total leakage signalexperiences the same scaling m(0) and rotation θ(0) as it traverses thesignal path so that vector C is transformed to vector C′ at thereceiver. Referring to FIG. 39, observe that vector C′ is the sum ofvector A′ and B′. Vector B′ is the vector that would have resulted ifvector B could have traversed the signal path by itself.

At the receiver, vector C′ is measured, e.g., by averaging the sampledcomplex signal captured from the output of the I/Q demodulator duringthe stimulation by vector B. Since both A′ and C′ are known bymeasurement, vector B′ can be calculated by subtraction. The DC scalingm(0) and the DC rotation θ(0) may be calculated from the vector B′ andthe vector B:

Map=m(0)exp(jθ(0))=B′/B.

Similarly, the inverse map that will undo the effect of the signal pathcan be determined from the inverse expression:

InverseMap=exp(−j(θ(0))/m(0)=B/B′.

Then the LO leakage vector A may be computed by multiplying vector A′ bythe inverse map. In practice, it may be advisable to keep the magnitudeof B to within an order of magnitude of A. It is also a good practice tonot transmit vector B by itself, but to transmit the sum of the vector Band another signal K, where the signal K has larger energy than thevector B signal and frequency content bounded away from DC since thetransmitter's LO leakage can potentially change with power in theinstantaneous bandwidth. For example, the signal K may be a tone.

In some embodiments, the sampled complex signals are windowed. If awindow is not applied, there are restrictions on the frequency of thetone K. In addition to tone K, if there are other signal tones present,they could also potentially leak into the measurement. Thus, if notusing a window, the tones (intentional or not) are preferably restrictedto certain frequencies to avoid leakage.

Method for Determining the Transmitter's LO Leakage

1. Stimulate the transmitter with a constant zero signal.

2. Measure the vector A′ produced at the receiver.

3. Stimulate the transmitter with non-zero complex constant B.

4. Measure vector C′ at the receiver.

5. Calculate the transmitter's LO leakage vector A from the followingequations:

B′=C′−A′  (1.64)

InvMap=B/B′  (1.65)

A=A′*InvMap.  (1.66)

Once the transmitter's LO leakage vector A has been computed, thetransmitter may remove (or substantially compensate for) the LO leakageby applying the translation vector −A to transmitted signals

I′(n)=I(n)−A _(I)

Q′(n)=Q(n)−A _(Q).

The compensation unit 3702 may apply this translation in addition to theI/Q impairment pre-compensation described above. For example, thecomplex signal (I(n),Q(n)) may be subjected to the 2×2 matrix of digitalfilters to pre-compensate for I/Q impairments, and then translated topre-compensate for the LO leakage.

In some embodiments, the calculation of the DC mapping may include thefollowing extra calculation. As described herein, the iterative methodmay diverge if the estimation error of the phase rotation is too large.In the case that the phase skew is large, this extra step may be used toget a more accurate estimate and enable the iterative method toconverge: (1) Calculate the mapping from RX to TX as is alreadydescribed. (2) Make a measurement of the phase skew. (3) Calculate usingthe method “Altering the gain imbalance and phase skew through a linearsystem” using the mapping from #1. (4) Add the rotation measurement of#1 to the computed phase skew of #3 to attain a more accurate rotationestimate.

Method for Computing a DC Mapping and DC Rotation for Signal Path

In one set of embodiments, a method 4000 may involve the actions shownin FIG. 40. The method 4000 may be used to estimate a DC scaling m(0) ofa signal path between an I/Q modulator of a transmitter and ademodulator of a receiver. (The method 4000 may incorporate any subsetof the features described above in the section “Calculating the MappingBetween the Rx and Tx”.) The method 4000 is described below as beingperformed by a “processing agent”. The processing agent may be anysystem of digital circuitry, e.g., a processor (executing under thecontrol of program instructions), a programmable hardware element, anASIC, or any combination thereof.

In some embodiments, the receiver conforms to a direct-conversionarchitecture, and the demodulator is an analog I/Q demodulator. In otherembodiments, the receiver may conform to a different architecture (e.g.,a superheterodyne architecture) which performs analog down-conversionfollowed by digital I/Q demodulation. Thus, in this case the demodulatoris realized by digital circuitry, e.g., on a programmable hardwareelement, in dedicated digital circuitry, in software on a processor, orany combination thereof.

At 4010, the processing agent may direct the transmitter to supply azero signal as input to the I/Q modulator. The zero signal is a constantzero signal. The zero signal may be a digital zero signal that issupplied to the complex input of the transmitter's DAC unit (e.g., DACunit 3705 of FIG. 37). Thus, I′(n)=0 and Q′(n)=0.

At 4015, the processing agent may receive a first response signal thathas been captured from the demodulator in response to the action ofsupplying the zero signal. The first response signal may be capturedfrom the output of the receiver's ADC unit. (See, e.g., digitizationunit 215 of FIG. 2B.)

At 4020, the processing agent may direct the transmitter to supply aconstant signal equal to a non-zero complex constant B=B₁+jB_(Q) asinput to the I/Q modulator. Again, the constant signal may be suppliedto the complex input of the transmitter's DAC unit. Thus, I′(n)=B_(I)and Q′(n)=B_(Q). In some embodiments, B is entirely real, i.e., B_(Q)=0.

At 4025, the processing agent may receive a second response signal thathas been captured from the demodulator in response to the action ofsupplying the constant signal. The second response signal may becaptured from the output of the receiver's ADC unit.

At 4030, the processing agent may average the first response signal toobtain a first average and averaging the second response signal toobtain a second average. The averaging helps to reduce noise in themeasurements.

At 4035, the processing agent may compute a difference between thesecond average and the first average, e.g., according to the expression:

Diff=SecondAvg−FirstAvg.

At 4040, the processing agent may compute the DC scaling based on thedifference and the non-zero complex constant, e.g., as described above.The processing agent may store the DC scaling in a memory.

In some embodiments, the method 4000 may also include computing a DCrotation θ(0) of the signal path based on a phase of the difference anda phase of the non-zero complex constant B, e.g., according to theexpression

θ(0)=Phase(Diff)/Phase(B).

In some embodiments, the DC scaling and DC rotation are used to removean effect of the signal path from measured I/Q impairments at thereceiver in order to obtain estimates of the I/Q impairments of thetransmitter.

In some embodiments, the signal path includes a cable coupling betweenthe transmitter and the receiver. In other embodiments, the signal pathincludes a wireless channel between the transmitter and the receiver.

As an alternative to computing a difference of averages, the processingagent may alternatively compute a difference signal by subtracting thefirst response signal from the second response signal, and then averagethe difference signal. The DC scaling may then be computed based onaverage value and the non-zero complex constant.

In one set of embodiments, a computer system for estimating a DC scalingm(0) of a signal path between an I/Q modulator of a transmitter and ademodulator of a receiver, the computer system comprising a processorand memory. The memory storing program instructions, where the programinstructions, when executed by the processor, cause the processor to:direct the transmitter to supply a zero signal as input to the I/Qmodulator; receive a first response signal that has been captured fromthe demodulator in response to said supplying the zero signal; directthe transmitter to supply a constant signal equal to a non-zero complexconstant as input to the I/Q modulator; receive a second response signalthat has been captured from the demodulator in response to saidsupplying the constant signal; average the first response signal toobtain a first average and averaging the second response signal toobtain a second average; compute a difference between the second averageand the first average; compute the DC scaling based on the differenceand the non-zero complex constant. The program instructions mayincorporate any subset of the features described above in the section“Calculating the Mapping Between the Rx and Tx” and in connection withmethod 4000.

Altering the Gain Imbalance and Phase Skew Through a Linear System

When calibrating the transmitter or measuring the I/Q impairments of thetransmitter, the method of this section may be used to remove theeffects of the signal path between the transmitter's I/Q modulator andthe receiver's I/Q demodulator from the receiver's I/Q impairmentmeasurements. Those effects may include the effects of the transmitter'sfront end, the transmission channel and the receiver's front end. Forexample, the transmitter's front end may include an RF filter thatcontributes to the frequency response of the signal path. Similarly, thereceiver's front end may include an RF filter that contributes to thefrequency response of the signal path.

In some embodiments, the magnitude response m(f) of the signal path maybe calibrated out, while the phase rotation θ(f) is not calibrated out.(Calibration may be achieved by performing pre-compensation at thetransmitter using digital circuit 510 and/or post-compensation at thereceiver using digital circuit 220.) The calculations of this sectionallow for the correct measurement of the transmitter's I/Q impairmentswithout first calibrating out the phase response of the signal path.

In some embodiments, the total frequency response (including bothmagnitude and phase rotation) of the signal is calibrated out.

Given a system with frequency response H(f) and an input signals_(input)(f,t) having gain imbalance g(f) and phase skew φ(f) as shownin FIG. 41, we will develop equations that allow us to determine thegain imbalance g′(f) and phase skew φ′(f) at the system outputs_(output)(f,t). We assume the input gain imbalance g(f) and input phaseskew φ(f) appear entirely on the Q input channel. However, we cannotsimultaneously make the same assumption at the system output. Ingeneral, the output components I′(t) and Q′(t) will have the form:

I′(t)=g _(I)(f)cos(2πft+φ _(I)(f))

Q′(t)=g _(Q)(f)sin(2πft+φ _(Q)(f)).

The output gain imbalance g′(f) and output phase skew φ′(f) may thendetermined by:

g′(f)=g _(Q)(f)/g _(I)(f)

φ′(f)=φ_(Q)(f)−φ_(I)(f).

The development starts with equations (1.60) through (1.62) given inFIG. 42, relying on the fact that s_(output)(f,t)=h(t)*s_(input)(f,t),where h(t) is the impulse response corresponding to H(f). Equations 1.61and 1.62 imply that:

g _(I)(f)exp(jφ _(I)(f))+g _(Q)(f)exp(jφ_(Q)(f))=H(f){1+g(f)exp(jφ(f))}  (1.63)

g _(I)(f)exp(−jφ _(I)(f))−g _(Q)(f)exp(−jφ_(Q)(f))=H(−f){1−g(f)exp(−jφ(f))}.  (1.64)

Define A(f) and B(f) to be respectively the right-hand sides of equation(1.63) and (1.64):

A(f)=H(f){1+g(f)exp(jφ(f))}  (1.65)

B(f)=H(−f){1−g(f)exp(−jφ(f))}.  (1.66)

Also, define w(f), x(f), y(f) and z(f) based on the left-hand sides ofequations (1.63) and (1.64):

g _(I)(f)exp(jφ _(I)(f))=w(f)+jx(f)  (1.67)

g _(Q)(f)exp(jφ _(Q)(f))=y(f)+jz(f).  (1.68)

It follows that

w(f)+jx(f)+y(f)+jz(f)=A(f)  (1.69)

w(f)−jx(f)−y(f)+jz(f)=B(f),  (1.70)

and thus,

w(f)=(½)Re{A(f)+B(f)}

x(f)=(½)Im{A(f)−B(f)}

y(f)=(½)Re{A(f)−B(f)}

z(f)=(½)Im{A(f)+B(f)}.

Note that if H(f) has an even magnitude response and an odd phaseresponse, i.e., H(−f)=H(f)*, then the impulse response h(t)corresponding to H(f) is entirely real. As a result, in this specificcase, the filter H(f) does not change the measurement of the I/Qimpairments:

A(f) + B(f) = H(f) + H^(*)(f) + g{H(f)exp (jϕ(f)) − H^(*)(f)exp (−jϕ(f))} = 2 Re(H) + 2 j gIm{H(f)exp (jϕ(f))}A(f) − B(f) = H(f) − H^(*)(f) + g{H(f)exp (jϕ(f)) + H^(*)(f)exp (−jϕ(f))} = 2j  Im{H(f)} + 2 gRe {H(f)exp (jϕ(f))}  w(f) = Re(H(f)), x(f) = Im(H(f))  g_(I)(f)exp (jϕ_(I)(f)) = w(f) + j x(f) = H(f)  y(f) = g Re{H(f)exp (jϕ(f))}  z(f) = g Im{H(f)exp (jϕ(f))}  g_(Q)(f)exp (jϕ_(Q)(f)) = y(f) + j z(f) = gH(f)exp (jϕ(f))  g_(Q)(f)exp (jϕ_(Q)(f))/g_(I)(f)exp (j ϕ_(I)(f)) = g exp (jϕ(f)).

The method below describes how to iteratively measure the TX impairmentswhen the magnitude and phase of the signal path transfer function H(f)are only approximately known. Part of that iterative measurement methodinvolves using the equations derived in this section to compute the I/Qimpairments at the output of the transmitter's I/Q modulator based onthe I/Q impairments at the input (or alternatively, at the output) ofthe receiver's I/Q demodulator. To perform this computation, thefrequency response H(f) is set equal to the inverse of an estimate ofthe frequency response of the signal path. Different estimates of thesignal path frequency response may be used in different circumstances.

Transforming I/Q Impairments Through a Linear System H(f)

In one set of embodiments, a method 4300 may involve the operationsshown in FIG. 43. The method 4300 may be used to compute I/Q impairmentsat a complex output z_(OUT) of an electrical system based on I/Qimpairments at a complex input z_(IN) of the electrical system. Acomplex input is an input that includes an inphase channel and aquadrature channel. Likewise a complex output is an output that includesan inphase channel and a quadrature channel. (Method 4300 may includeany subset of the features described above in the section “Altering theGain Imbalance and Phase Skew through a Linear System”.) Method 4300 maybe performed by a processing agent as described above.

At 4310, the processing agent may compute a spectrum A(f) according tothe expression

H(f){1+g(f)exp(jφ(f))},

where H(f) is a spectrum of a linear system model of the electricalsystem, where g(f) is a gain imbalance at the complex input z_(IN),where φ(f) is a phase skew at the complex input z_(IN).

At 4315, the processing agent may compute a spectrum B(f) according tothe expression

H(−f){1−g(f)exp(−jφ(f))}.

At 4320, the processing agent may compute a sum of the spectra A(f) andB(f), and a difference of the spectra A(f) and B(f), e.g., according tothe relations:

Sum(f)=A(f)+B(f),

Diff(f)=A(f)−B(f).

At 4325, the processing agent may compute a gain imbalance and phaseskew at the complex output Z_(OUT) based on real and imaginary parts ofthe sum, and real and imaginary parts of the difference. In particular,as described above, the functions g_(I)(f), g_(Q)(f), φ_(I)(f) andφ_(Q)(f) may be computed based on the sum spectrum and the differencespectrum, and then the gain imbalance and phase skew at the complexoutput z_(OUT) may be computed based on g_(I)(f), g_(Q)(f), φ_(I)(f) andφ_(Q)(f) as shown in FIG. 41. The output gain imbalance and output phaseskew constitute useful information in part because they may be used toperform I/Q impairment compensation or calibration as variouslydescribed herein.

The processing agent may store the output gain imbalance and the outputphase skew in a memory.

In some embodiments, the electrical system being modeled by the spectrumH(f) is the inverse of a signal path from an I/Q modulator of atransmitter to a demodulator of a receiver, e.g., as variously describedherein. The gain imbalance and the phase skew at the complex inputz_(IN) of the electrical system may represent a gain imbalance and aphase skew at the input (or alternatively, at the output) of thedemodulator. The gain imbalance and the phase skew at the complex outputz_(OUT) of the electrical system may represent a gain imbalance and aphase skew at the output of the I/Q modulator.

In some embodiments, the receiver conforms to a direct-conversionarchitecture, and the demodulator is an analog I/Q demodulator. In otherembodiments, the receiver may conform to a different architecture (e.g.,a superheterodyne architecture) which performs analog down-conversionfollowed by digital I/Q demodulation. Thus, in this case the demodulatoris realized by digital circuitry, e.g., on a programmable hardwareelement, in dedicated digital circuitry, in software on a processor, orany combination thereof.

In some embodiments, the processing agent may also include computing aninverse of a spectrum of the signal path to determine the spectrum H(f),e.g., as variously described herein.

In some embodiments, the spectrum H(f) may be determined (or estimated)based on a DC scaling and a DC rotation of the signal path, e.g.,according to the relation

H(f)=exp{−jθ(0)}/m(0).

In some embodiments, the processing agent may compute the DC scaling andthe DC rotation by: supplying a zero signal as input to the I/Qmodulator, capturing a first response signal from the I/Q demodulator inresponse to said supplying the zero signal; supplying a constant signalequal to a non-zero complex constant as input to the I/Q modulator;capturing a second response signal from the I/Q demodulator in responseto said supplying the constant signal; averaging the first responsesignal to obtain a first average and averaging the second responsesignal to obtain a second average; computing a difference between thesecond average and the first average; and computing the DC scaling basedon the difference and the non-zero complex constant.

In some embodiments, the processing agent may also measure (e.g., directthe measurement of) the gain imbalance g(f) and the phase skew φ(f) ofan electronic device at a plurality of frequencies. The electronicdevice may be a transmitter, a receiver, or the series combination of atransmitter and receiver, as variously described herein.

In some embodiments, the processing agent may be a programmable hardwareelement. In other embodiments, the processing agent may be a processorthat is configured to perform the method 4300 in response to executionof program instructions.

Determination of Transmitter I/Q Impairments Using Shared LOs

In one set of embodiments, a method 4400 for determining I/Q impairmentsof a transmitter may involve the actions shown in FIG. 44. (Furthermore,the method 4400 may include any subset of the features described in thesection “Iterative Technique for Measuring Tx Impairments, in thesection “Iterative Estimation of Transmitter Impairments Using SharedLOs”, and in the section “Iterative Estimation of Transmitter ImpairmentUsing Shared LO—Optimized”.) The method 4400 may be enacted by aprocessing agent, e.g., a processing agent as variously described above.

At 4410, the processing agent may perform a set of operations. The setof operations may include the operations 4415 through 4440, as shown inFIG. 44.

At 4415, the processing agent may direct that a complex exponential toneat frequency f be supplied to the transmitter. For example, theprocessing agent may issue commands causing the complex exponential toneto be supplied (or generated by) the transmitter. The frequency f may beinterpreted as a displacement frequency relative to the transmitter'slocal oscillator frequency. The frequency f may be non-zero.

At 4420, the processing agent may supply a pre-compensationtransformation to a pre-compensation circuit of the transmitter. Thepre-compensation circuit may be configured to apply the pre-compensationtransformation to the complex exponential tone to obtain an adjustedcomplex signal. (For example, the pre-compensation circuit may be thedigital circuit 510 of FIG. 5 or the compensation unit 3702 of FIG. 37.)The pre-compensation transformation may be configured to pre-compensatefor a current estimate of the I/Q impairments of the transmitter. Thetransmitter may be configured to transmit a transmit signal based on theadjusted complex signal, e.g., as variously described above. Thereceiver may be configured to receive the transmit signal and capture asampled complex signal representing the received transmit signal, e.g.,as variously described above. (The action of “sampling” a complex signalinvolves sampling its I component and sampling its Q component. Thus, a“sampled complex signal” includes a sampled I signal and a sampled Qsignal.)

At 4425, the processing agent may compute raw I/Q impairments based onthe sampled complex signal. For example, the raw I/Q impairments mayinclude a gain imbalance and phase skew of the sampled complex signal.See the section “Precise Measurement Technique” for information on howto compute the raw I/Q impairments.

At 4430, the processing agent may transform the raw I/Q impairments todetermine transformed I/Q impairments. The transform may remove measuredI/Q impairments of the receiver from the raw I/Q impairments. See thesection “Removing Receiver Impairments from Measured Output Impairments”for more information on how perform this transform.

As an alternative to operations 4425 and 4430, the processing agent mayapply a 2×2 matrix of digital filters to the sampled complex signal toremove the effect of the receiver's measured I/Q impairments, e.g., asdescribed above in connection with FIGS. 2A, 2B and 3, and in thesections “Wideband I/Q Impairment Equalization” and “Filter Design”. Theapplication of the 2×2 matrix of digital filters to the sampled complexsignal produces a filtered complex signal. The filtered complex signalmay be used to compute the transformed I/Q impairments. The methoddescribed in the section “Precise Measurement Technique” may be used todetermine the transformed I/Q impairments based on the filtered complexsignal.

At 4435, the processing agent may remove a current estimate of a signalpath from the transformed I/Q impairments to obtain path-compensated I/Qimpairments, where the signal path includes a path from an I/Q modulatorof the transmitter to a demodulator of the receiver. (The signal pathestimate may be removed by using the method described in the section“Altering the Gain Imbalance and Phase Skew Through a Linear System”.)The path-compensated I/Q impairments may represent an estimate forresidual I/Q impairments of the transmitter, i.e., “residual” in thesense that they are remaining impairments after the partial correctionrealized by the pre-compensation transformation of 4420.

In some embodiments, the receiver may conform to a direct-conversionarchitecture, and the demodulator may be an analog I/Q demodulator, inwhich case the sampled complex signal may be captured by digitizing thecomplex analog output of the analog I/Q demodulator. In otherembodiments, the receiver may conform to a different kind ofarchitecture, e.g., a superheterodyne architecture. Thus, the receivermay generate a real analog signal (e.g., a real intermediate-frequencysignal) that represents the received transmit signal. The real analogsignal may be digitized to obtain a sampled real signal. The sampledcomplex signal may then be generated computationally, e.g., by digitallymixing the sampled real signal with an orthogonal pair of digitalsinusoids to obtain respectively the I and Q components of the sampledcomplex signal.

At 4440, the processing agent may update the current estimate of the I/Qimpairments of the transmitter based on the path-compensated I/Qimpairments, e.g., by combining the path-compensated I/Q impairmentswith the respective impairments of the current estimate.

In some embodiments, the method 4400 may include repeating the set ofoperations to determine a converged estimate (stable estimate) of theI/Q impairments of the transmitter at the frequency f. (This convergedestimate comprises a measurement of the transmitter's I/Q impairments atthe frequency f.) The set of operations may be repeated until a qualitymeasure based on the path-compensated I/Q impairments is larger than athreshold. The converged estimate may be used to at least partiallycompensate for the I/Q impairments of the transmitter at frequency f,e.g., as variously described herein.

In some embodiments, the above-described action of repeating the set ofoperations may itself be performed a plurality of times to determine theconverged estimate at a plurality of different values for the frequencyf. The above-described action of repeating the set of operations todetermine the converged estimate at frequency f may be referred toherein as a “transmitter I/Q impairment measurement at frequency f”.Thus, a plurality of transmitter I/Q impairment measurements may be madeso as to cover the plurality of frequency values.

In some embodiments, the plurality of frequency values are symmetricabout zero. Furthermore, the transmitter I/Q impairment measurements maybe made so that the frequency values are visited in manner thatalternates in sign and is non-decreasing in absolute value, e.g., asvariously described herein.

In some embodiments, a local oscillator of the transmitter and a localoscillator of the receiver are phase locked to the same frequencyreference (infers frequency lock).

In some embodiments, the current estimate of the signal path is based ona DC scaling and a DC rotation of the signal path, at least for a firstof the transmitter I/Q impairment measurements.

In some embodiments, the DC scaling and the DC rotation may bedetermined by: supplying a zero vector signal to the transmitter;supplying a non-zero DC vector signal to the transmitter; and computingthe DC scaling and the DC rotation based on a first DC vector responseand a second DC vector response, where the first DC vector response ismeasured at the receiver in response to the zero vector signal, wherethe second DC vector response is measured at the receiver in response tothe non-zero DC vector signal. For more information on how to computethe DC scaling and DC rotation, see the section “Calculating the MappingBetween RX and TX”.

In some embodiments, the pre-compensation transformation has the form ofa 2×2 matrix, where at least a first diagonal element of the matrix iscomputed from the current estimate of the I/Q impairments of thetransmitter at frequencies f and −f, and where at least a firstnon-diagonal element of the matrix is computed from the current estimateof the I/Q impairments of the transmitter at frequencies f and −f.

In some embodiments, the current estimate of the signal path includes ameasured amplitude of the sampled complex signal at frequency f. Theamplitude may be measured as described in the section “PreciseMeasurement Technique”.

In some embodiments, the current estimate of the signal path alsoincludes a measured rotation of the sampled complex signal at frequencyf.

Determination of Transmitter I/Q Impairments with Offset LOs

In one set of embodiments, a method 4500 for determining I/Q impairmentsof a transmitter may involve the actions shown in FIG. 45. (Furthermore,the method 4500 may include any subset of the features described in thesection “Iterative Techniques for Measuring Tx Impairments.) The method4500 may be performed by a processing agent, e.g., a processing agent asvariously described above.

At 4510, the processing agent may configure a local oscillator (LO) ofthe transmitter and a local oscillator (LO) of the receiver to be phaselocked to a common reference and so that a frequency of the receiver'sLO minus a frequency of the transmitter's LO frequency is equal to anon-zero amount ΔLO. The amount ΔLO may be positive or negative.

At 4520, the processing agent may perform a set of operations S_(O). Theset S_(O) may include operations 4525 through 4550, as shown in FIG. 45.

At 4525, the processing agent may direct that a complex exponential toneat frequency f be supplied to the transmitter. (The frequency f may beinterpreted as a displacement relative to the transmitter's LOfrequency.) The complex exponential tone may be supplied in digitalform, e.g., as variously described above. In some embodiments, thetransmitter may couple to (or include) a programmable hardware elementconfigured to generate the complex exponential tone. To facilitate thisgeneration, the PHE may receive the sample clock used by thetransmitter's DAC unit.

At 4530, the processing agent may supply a pre-compensationtransformation to a pre-compensation circuit of the transmitter. Thepre-compensation circuit may be configured to apply the pre-compensationtransformation to the complex exponential tone in order to obtain anadjusted complex signal. (For example, the pre-compensation circuit maybe the digital circuit 510 of FIG. 5 or the compensation unit 3702 ofFIG. 37.) The pre-compensation transformation may be configured topre-compensate for a current estimate of the I/Q impairments of thetransmitter. The transmitter may be configured to transmit a transmitsignal based on (derived from) the adjusted complex signal, e.g., asvariously described above. A receiver may be configured to receive thetransmit signal and to capture a sampled complex signal representing thereceived transmit signal, e.g., as variously described above. Thetransmitter may transmit the transmit signal onto a transmission channel(e.g., a cable), and the receiver may receive the transmit signal fromthe channel.

At 4535, the processing agent may frequency shift the sampled complexsignal by the amount ΔLO to obtain a frequency-shifted signal, e.g., bymultiplying the sampled complex signal by a discrete-time complexexponential signal running at frequency ΔLO.

At 4540, the processing agent may compute raw I/Q impairments atfrequency f based on the frequency-shifted signal. The raw I/Qimpairments may include a gain imbalance g_(R)(f) and a phase skewφ_(R)(f). (The process of computing the I/Q impairments from a complexsignal is described above.)

At 4545, the processing agent may remove a current estimate of a signalpath from the raw I/Q impairments at frequency f to obtainpath-compensated I/Q impairments at frequency f (e.g., as describedabove in the section “Transforming I/Q Impairments through a LinearSystem”, or in the section “Altering the Gain Imbalance and Phase Skewthrough a Linear System”). The signal path may include a path from anI/Q modulator of the transmitter to a demodulator of the receiver. Thepath-compensated I/Q impairments at frequency f may represent anestimate for residual I/Q impairments of the transmitter at frequency f.

In some embodiments, the receiver may conform to a direct-conversionarchitecture, and the demodulator may be an analog I/Q demodulator, inwhich case the sampled complex signal may be captured by digitizing thecomplex analog output of the analog I/Q demodulator. In otherembodiments, the receiver may conform to a different kind ofarchitecture, e.g., a superheterodyne architecture. Thus, the receivermay generate a real analog signal (e.g., a real intermediate-frequencysignal) that represents the received transmit signal. The real analogsignal may be digitized to obtain a sampled real signal. The sampledcomplex signal may then be generated computationally, e.g., by digitallymixing the sampled real signal with an orthogonal pair of digitalsinusoids to obtain respectively the I and Q components of the sampledcomplex signal.

At 4550, the processing agent may update the current estimate of the I/Qimpairments of the transmitter at frequency f based on thepath-compensated I/Q impairments at frequency f.

In some embodiments, the method 4500 may include repeating the set ofoperations S_(O) to determine a converged estimate (or stable estimate)of the I/Q impairments of the transmitter at frequency f. (Thisconverged estimate may be interpreted as a measurement of thetransmitter's I/Q impairments at frequency f.) For example, the set ofoperations may be repeated until a quality measure based on thepath-compensated I/Q impairments is larger than a threshold. (Thequality measure may be the negative of image rejection at frequency f.)The converged estimate is usable to at least partially compensate forthe I/Q impairments of the transmitter at frequency f. Theabove-described action of frequency shifting may be performed using afrequency-shift signal that is phase continuous between successiverepetitions of the set of operations.

In some embodiments, the method 4500 may also include performing saidrepeating (of the set of operations S_(O)) a plurality of times todetermine the converged estimate at a plurality of different values forthe frequency f, e.g., values covering a desired transmission (orcommunication) band.

In some embodiments, the set of operations S_(o) may also includeremoving measured I/Q impairments of the receiver at frequency f−ΔLOfrom the sampled complex signal prior to the frequency shiftingoperation. The measured I/Q impairments of the receiver at frequencyf−ΔLO may be removed by multiplying the sampled complex signal by a 2×2matrix M=(m_(ij)) of constants, e.g., according to the relations:

${\begin{bmatrix}{I^{\prime}(n)} \\{Q^{\prime}(n)}\end{bmatrix} = {\begin{bmatrix}m_{11} & m_{12} \\m_{21} & m_{22}\end{bmatrix}\begin{bmatrix}{I(n)} \\{Q(n)}\end{bmatrix}}},$

where I(n) and Q(n) denote respectively the inphase and quadraturecomponents of the sampled complex signal. In one embodiment, the matrixM may have the special form

${M = \begin{bmatrix}1 & 0 \\m_{21} & m_{22}\end{bmatrix}},$

and the constants m₂₁ and m₂₂ may be determined from a receiver's gainimbalance g_(RX)(f−ΔLO) and receiver's phase skew φ_(RX)(f−ΔLO) atfrequency f−ΔLO based on the expressions:

$m_{21} = {{{- \tan}{\left\{ {\phi_{RX}\left( {f - {\Delta \; {LO}}} \right)} \right\}.m_{22}}} = {\frac{1}{{g_{RX}\left( {f - {\Delta \; {LO}}} \right)}\cos \left\{ {\phi_{RX}\left( {f - {\Delta \; {LO}}} \right)} \right\}}.}}$

See the section entitled “Performing Traditional Impairment Compensationat a Single Frequency”.

In an alternative embodiment, the constants m₂₁ and m₂₂ may bedetermined based on the receiver's measured I/Q impairments at frequencyf−ΔLO and its negative −(f−ΔLO), as described in the section“Calculating True Single Point Vector Calibration Constants”, andespecially at equations (1.81) and (1.82).

In some embodiments, the receiver's I/Q impairments may be measured aspart of method 4500, i.e., measured based on the sampled complex signalprior to frequency shifting. For example, the set of operations S_(o)may include measuring the I/Q impairments of the receiver at frequencyf−ΔLO based on the sampled complex signal. One technique for performingthis measurement involves: (a) computing a Discrete-Time Fouriertransform value C_(I) at frequency f−ΔLO of an I component of thesampled complex signal; (b) computing a Discrete-Time Fourier transformvalue C_(Q) at frequency f−ΔLO of a Q component of the sampled complexsignal; (c) computing the receiver gain imbalance at frequency f−ΔLObased on the magnitudes of the values C_(I) and C_(Q); and (d) computingthe receiver phase skew at frequency f−ΔLO based on the phases of thevalues C_(I) and C_(Q). For more information on embodiments of thistechnique, see the “Precise Measurement Technique” section.

In some embodiments, the method 4500 may also include applying atime-domain window to the sampled complex signal prior to computing thevalues C_(I) and C_(Q). The time-domain window may be a rectangular(uniform) window or any of a variety of standard non-uniform windows.For more information on use of the rectangle window, see the section“Rectangle Window Optimization”.

In some embodiments, the above-described measurement of the receiver'sI/Q impairments and the estimation of the transmitter's I/Q impairmentsmay be performed at least partially in parallel. For example, in oneembodiment, a programmable hardware element (or perhaps a multicoreprocessor) may be configured to perform the measurement of thereceiver's I/Q impairments in parallel with the frequency shiftoperation on the sampled complex signal.

In some embodiments, the set of operations may include measuring thereceiver's I/Q impairments at frequency f−ΔLO as described above,computing the 2×2 matrix of correction constants based on the measuredI/Q impairments as described above, and then applying the 2×2 matrix tothe sampled complex signal prior to the frequency shift operation. Inother words, the frequency shift operation is applied to the modifiedcomplex signal (I′(n),Q′(n)) resulting from the application of the 2×2matrix.

In some embodiments, it is assumed that the I/Q impairments of thereceiver have already been measured over the frequency band of interestprior to the execution of method 4500. Thus, a 2×2 matrix of digitalfilters may be designed based on the receiver's I/Q impairments, asdescribed above in connection with FIGS. 2A, 2B and 3, and in thesections “Wideband I/Q Impairment Equalization” and “Filter Design”. Theset of operations may include the operation of applying the 2×2 matrixof digital filters to the sampled complex signal prior to the frequencyshifting operation. The resulting filtered complex signal may then besubjected to the frequency shifting.

In some embodiments, the pre-compensation transformation has the form ofa 2×2 matrix, and the matrix has the property that at least one of thediagonal elements of the matrix is computed based on the currentestimate of the I/Q impairments of the transmitter at frequency f and acurrent estimate of the I/Q impairments of the transmitter at frequency−f, and the property that at least one of the non-diagonal elements ofthe matrix is computed based on the current estimate of the I/Qimpairments of the transmitter at frequency f and the current estimateof the I/Q impairments of the transmitter at frequency −f. In someembodiments, each of the four matrix elements is computed in thisfashion.

As described above, the processing agent may remove a current estimateof a signal path from the raw I/Q impairments at frequency f to obtainpath-compensated I/Q impairments at frequency f. In some embodiments,the current estimate of the signal path may include a measured amplitudeof the frequency-shifted signal at frequency f. In one embodiment, thecurrent estimate of the signal path may also include a measured rotationof the frequency-shifted signal at frequency f.

In some embodiments, the current estimate of the signal path may bebased on a DC scaling and a DC rotation of the signal path. Such anestimate may be used in at least a first performance of said set ofoperations.

In some embodiments, the method 4500 may also include determining the DCscaling and the DC rotation by: supplying a zero vector signal to thetransmitter; supplying a non-zero DC vector signal to the transmitter;computing the DC scaling and the DC rotation based on a first DC vectorresponse and a second DC vector response, where the first DC vectorresponse is measured at the receiver in response to the zero vectorsignal, where the second DC vector response is measured at the receiverin response to the non-zero DC vector signal. For more information ondetermination of the DC scaling and DC rotation, see the section“Calculating the Mapping Between the Rx and Tx” and the section “Methodfor Computing a DC Mapping and DC Rotation for Signal Path”.

Determining I/Q Impairments of a Receiver

In one set of embodiments, a method 4600 for determining I/Q impairmentsof a receiver may include the operations shown in FIG. 46. The method4600 may be performed by a processing agent as described above.

At 4610, the processing agent may direct that an input signal besupplied to the receiver. In other words, the processing agent may issuecommands to cause the input signal to be supplied to (or generated by)the receiver. The input signal may include an isolated tone atdisplacement frequency f and a void interval (i.e., an intervalcontaining only noise) around displacement frequency −f (To say that atone is “isolated” at a given frequency means that the tone is the onlyenergy source except for noise in a frequency neighborhood of the givenfrequency (e.g., in an interval of frequencies centered on the givenfrequency). If the noise energy is too large, the measurement qualitywill degrade. The tone is preferably the only significant source ofenergy in the frequency neighborhood.) The receiver may be configured todemodulate the input signal in order to obtain a sampled complex signal,e.g., as variously described above. The displacement frequencies f and−f may be displacements relative to a local oscillator frequency of thereceiver.

At 4615, the processing agent may compute the I/Q impairments of thereceiver at frequency f based on the sampled complex signal.

At 4620, the processing agent may repeat the actions of directing (4610)and computing (4615) for values of the frequency f spanning a specifiedfrequency band, e.g., the currently-selected input band of the receiveror a standardized communication band.

At 4625, the processing agent may store the receiver's I/Q impairmentsfor each of the values of the frequency f in a memory.

In some embodiments, the input signal is supplied by a transmitter whoselocal oscillator frequency is offset by a non-zero value from the localoscillator frequency of the receiver, e.g., as variously describedabove.

In some embodiments, the input signal is supplied by a calibration tonesynthesizer. A calibration tone synthesizer is a system configured tocreate quality tones for the purpose of calibrating other systems. Insome embodiments, the term “quality tone” implies stability overamplitude, frequency, temperature, or time. In one embodiment, thereceiver includes a calibration tone synthesizer to facilitate selfcalibration.

In some embodiments, the action of computing the I/Q impairments of thereceiver at frequency f includes: computing a Discrete-Time Fouriertransform value C_(I) at frequency f of an I component of the sampledcomplex signal; computing a Discrete-Time Fourier transform value C_(Q)at frequency f of a Q component of the sampled complex signal; computinga gain imbalance of the receiver at frequency f based on magnitudes ofthe values C_(I) and C_(Q); and computing a phase skew of the receiverat frequency f based on phases of the values C_(I) and C_(Q).

In some embodiments, the method 4600 may also include applying atime-domain window to the sampled complex signal prior to said computingof the values C_(I) and C_(Q), e.g., as describe below in the section“Precise Measurement Technique”.

Measuring I/Q Impairments Associated with Complex Signal

In one set of embodiments, a method 4700 may include the operationsshown in FIG. 47. The method 4700 may be used to measure I/Q impairmentsassociated with a sampled complex signal produced by a receiver. Themethod 4600 may be performed by a processing agent (e.g., a computersystem executing under the control of program instructions).

At 4710, the processing agent may direct a device to stimulate thereceiver with a stimulus signal having an isolated tone at displacementfrequency f and a void interval at displacement frequency −f. Thedisplacement frequencies f and −f may be interpreted as displacementswith respect to a local oscillator frequency of the receiver. Thesampled complex signal may be a baseband signal produced by the receiverin response the action of stimulating with the stimulus signal.

At 4715, the processing agent may compute a Discrete-Time Fouriertransform value C_(I) at frequency f of an I component of the sampledcomplex signal;

At 4720, the processing agent may compute a Discrete-Time Fouriertransform value C_(Q) at frequency f of a Q component of the sampledcomplex signal.

At 4725, the processing agent may compute a gain imbalance g of thesampled complex signal at frequency f based on magnitudes of the valuesC_(I) and C_(Q), where the gain imbalance g includes a gain imbalance ofthe receiver.

At 4730, the processing agent may compute a phase skew φ of the sampledcomplex signal at frequency f based on phases of the values C_(I) andC_(Q), where the phase skew φ includes a phase skew of the receiver.

In some embodiments, the processing agent may apply a time-domain windowto the sampled complex signal prior to said computing of the valuesC_(I) and C_(Q).

In some embodiments, the device that provides the input signal is acalibration tone generator.

In some embodiments, the device is a transmitter whose local oscillatorfrequency is (intentionally) offset from the receiver's local oscillatorfrequency by a non-zero amount. In one such embodiment, the sampledcomplex signal has been subjected to frequency shifting to remove adifference between the local oscillator frequencies, in which case thegain imbalance g and phase skew φ may be partially dependent on thetransmitter's I/Q impairments. In particular, the gain imbalance g andthe phase skew φ may represent a composite effect of the transmitter'sI/Q impairments, the distortion introduced by the signal path (betweenthe transmitter's I/Q modulator and the receiver's demodulator), and thereceiver's I/Q impairments. In another such embodiment, the sampledcomplex signal is a raw signal from the demodulator that has not beensubjected to the above-described frequency shifting, and thus, the gainimbalance g and the phase skew φ may be interpreted as including onlyimpairments introduced by the receiver.

In some embodiments, the method 4700 may also include applying atime-domain window to the sampled complex signal prior to computing thevalue C_(I) and computing the value C_(Q), e.g., as described below.

In some embodiments, the receiver is a vector signal analyzer.

In some embodiments, one or more of the operations 4715 through 4730 maybe performed by a programmable hardware element.

In some embodiments, one or more of the operations 4715 through 4730 maybe performed in dedicated digital circuitry.

In some embodiments, one or more of the operations 4715 through 4730 maybe performed by a processor in response to the execution of programinstructions.

Offset LO Calibration Technique

The offset local oscillator (LO) method allows for I/Q impairmentmeasurement and carrier leakage measurement of both the receiver (RX)and the transmitter (TX) simultaneously. This method uses independentlytunable LOs for the transmitter and receiver, e.g., as shown in FIG. 48.In some embodiments, the step size of the transmitter LO and/or the stepsize of the receiver LO can be fractional or integer in nature. In someembodiments, the step size of the transmitter and/or the step size ofthe receiver LO should be a small percentage of the total instantaneousbandwidth.

The transmitter includes an I/Q modulator 4810 and a front end 4815. Acomplex exponential tone at non-zero displacement frequency f isprovided to the I/Q modulator 4810. The I/O modulator 4810 modulates acarrier signal (also referred to as “local oscillator signal”) with thetone to obtain a modulated signal. The carrier signal is provided by thetransmitter LO 4805. The modulated signal is transmitted onto atransmission medium (e.g., a cable 4820) by the transmitter front end4815.

The front end 4830 of the receiver receives the transmitted signal andconditions the received signal to obtain a conditioned signal. The I/Qdemodulator 4835 demodulates the conditioned signal using a carriersignal provided by the receiver LO 4840, resulting in a demodulatedsignal having components denoted as RX I and RX Q.

As shown in FIG. 49, shifting the RX and TX carriers off of each othercauses the tone, the receiver's image of the tone, the transmitter'simage of the tone, the transmitter's carrier leakage and the receiver'scarrier leakage to appear at different frequencies. The illustratedspectrum is based on the demodulated signal at the receiver. Thetransmitter produces the tone at 31 MHz. The spectrum includes twodifferent carrier leakages, one due to the transmitter's LO leakage andone due to the receiver's LO leakage. The spectrum also includes twodifferent primary images of the tone, one due to the transmitter's I/Qimpairments and one due to the receiver's I/Q impairments. Additionally,the spectrum includes the receiver's image of the transmitter's image,and the receiver's image of the transmitter's carrier leakage, both dueto the receiver's I/Q impairments. In this example the receiver'scarrier is placed 6 MHz below that of the transmitter's carrier. Thismakes the tone, the transmitter's image and the transmitter's leakageappear 6 MHz higher in frequency at the receiver than at thetransmitter. Then, as a result of the I/Q demodulator's impairments,each of these three signals (tone, TX image, and TX leakage) created bythe transmitter has a corresponding image after the I/Q demodulator, inaddition to the receiver's leakage.

Knowing the frequency offset between the transmit and receive LOs andthe frequency of the tone produced at the transmitter before themodulator, the exact spectral locations of all impairment artifacts canbe completely determined. If we let

FreqOffset=TxCarrierFrequency−RxCarrierFrequency,  (1.75)

the frequency locations (as seen by the receiver) of the spectralfeatures in the received spectrum are:

$\begin{matrix}{{RxTone} = {{TxTone} + {FreqOffset}}} & (1.76) \\{{TxLeakage} = {FreqOffset}} & (1.77) \\{{TxImage} = {{FreqOffset} - {TxTone}}} & (1.78) \\{{RxImage} = {{- {TxTone}} - {FreqOffset}}} & (1.79) \\{{RxLeakage} = {0\mspace{14mu} {Hz}}} & (1.80) \\\begin{matrix}{{RxImageofTxImage} = {{TxTone} - {FreqOffset}}} \\{= {{RxTone} - {2\; {FreqOffset}}}}\end{matrix} & (1.81) \\{{RxImageOfTxLeakage} = {- {{FreqOffset}.}}} & (1.82)\end{matrix}$

Measuring the receiver's I/Q impairments and carrier leakage isperformed in the same way as done in the “Precise Measurement Technique”section. However, measuring the transmitter's impairments is generallymore involved since there are multiple things to consider. Measuring thetransmitter's impairments may involve removing the receiver impairments.FIG. 50 shows the received spectrum after removal of the receiver's I/Qimpairments. After that removal, the spectrum may be frequency shiftedby −FreqOffset, as shown in FIG. 51. Now the frequency location of the“tone” in the shifted spectrum is the same as the frequency f at whichthe tone was originally produced at the transmitter. In addition, thetransmitter's leakage (TX leakage) and transmitter's image (TX image)are at the correct frequency locations (−f and zero, respectively) touse the algorithm found in the “Precise Measurement Technique” sectiononce the rotation is calculated and removed. (The rotation may becalculated using the method described in the section “Calculating theMapping Between and RX and TX”.) This algorithm will give an estimatefor the transmitter's I/Q impairments and the transmitter's LO leakagevector. This method for measuring the transmitter's I/Q impairments willwork as long as the signal path (including the transmitter's front endand the receiver's front end) has an even magnitude response and oddphase response. In reality, this is not the case, and even smallperturbations in magnitude or phase cause serious problems for themeasurements. The iterative algorithm removes this issue. The iterationsof the iterative algorithm involve performing pre-correction (e.g.,using the method of the section “Calculating True Single Point VectorCalibration Constants) based on the current estimate of thetransmitter's impairments, and removing the best available estimate ofthe signal path from the impairments measured at the receiver (using themethod of the section “Altering the Gain Imbalance and Phase SkewThrough a Linear System”). The iterative algorithm allows thetransmitter impairments to be measured even though there is error in theinitial estimate of those impairments.

Measuring the transmitter's impairments can be optimized one furtherstep by doing everything in the method above except removing thereceiver's impairments. Shown in FIG. 52 is the frequency-shiftedspectrum without first removing the receiver's impairments. By leavingthese impairments in the spectrum, the measured impairment values atfrequency f (i.e., 31 MHz in this example) are not exactly equal to thetransmitter's I/Q impairments since the receiver's impairments havedistorted the measurements. However, the same iterative algorithm usedto remove distortions of the RF front ends can also remove thedistortions due to the receiver's impairments. While ideally it isbetter to remove the receiver's impairments, in practice this takesextra time during calibration.

Restrictions

While this method is highly desirable in that multiple measurements canbe made in parallel, it does come with restrictions. The primaryrestriction is that is cannot be used to measure amplitude as itmeasures the combination of receiver amplitude and transmitter amplitudewithout any way to separate the two without another measurement.However, if either the receiver amplitude or the transmitter amplitudeare known, then the two can be separated. In most cases, the amplitudechanges slower with respect to frequency than the I/Q impairments.Therefore, a separate measurement procedure can be used to measureeither receiver amplitude or transmitter amplitude with a coarserfrequency step size than the step size used to determine the I/Qimpairments over the instantaneous bandwidth. As a result, the totalmeasurement time including amplitude is still much faster than thealternative.

Another minor problem with the offset LO method is that it placesrestrictions on the calibration frequency plan. Depending on the valueof the LO offset ΔLO, it is possible to get corrupt measurements atvarious measurement offsets. As shown in FIG. 49, there are sevenlocations in the frequency spectrum where energy appears in response tothe transmission of the tone. In order to correctly measure all of theimpairments for both the transmitter and the receiver, all seven ofthese signals must remain orthogonal, i.e. no two of the signals cannotoccur at the same frequency location. For example, if the receiver's LOis set to 2.400 GHz and the transmitter's LO is set to 2.396 GHz, thenmeasurement corruptions will occur when the transmitted baseband toneis: at 4 MHz since this would place the tone at exactly the RX leakage;at −4 MHz since this would place the TX image at the RX leakage; or at 8MHz since this would place the RX image at the TX leakage. In order toavoid these problems, the transmitted tone (TxTone) cannot be located atthe frequencies

{N*FreqOffset: N=−3,−2,−1,0,1,2,3}.

In addition, there are bandwidth limitations. The total measureablebandwidth is (TotalBW−LO_StepSize), while the total symmetricmeasureable bandwidth is (TotalBW−2*LO_StepSize). This is why the LOstep size must be a fraction of the total instantaneous bandwidth,preferably a small fraction. For example, if the instantaneous bandwidthis 100 MHz, and the LO step size is only 25 MHz, then 75 MHz of thebandwidth is theoretically measureable. In reality, since we typicallydesire symmetric bandwidth (i.e. +/−25 MHz rather than −25 MHz-50 MHz),our symmetric measureable bandwidth is only 50 MHz. In addition, thereis also less measureable bandwidth due to roll-off effects atband-edges.

Calculating True Single Point Vector Calibration Constants

This section shows how to calculate constants for a true single-pointcalibration that will perfectly pre-correct at a single location (i.e.,perfectly pre-compensate for I/Q impairments at a single frequency f)given knowledge of the I/Q impairments at f and −f, as indicated inFIGS. 53A and 53B. The single-point vector calibration correction 5310precedes a double-point vector corruption model 5320. Thus, a complexexponential tone at frequency f, which is provided as input to thesingle-point vector calibration correction, is pre-distorted to producea complex signal

cos(2πft)+jΓ sin(2πft+θ).

The pre-distorted signal is further distorted by the corruption model5320, resulting in a corrected output signal that is equal to theoriginal complex exponential tone.

From the “Corruption I/Q Impairments” section, we know how to develop a2×2 frequency response matrix H that represents the I/Q impairments of asystem. In that section, we found that A(f), E_(B)(f), C(f) and D(f) aredetermined by the “double point I/Q impairments”, i.e., by the I/Qimpairments at f and the I/Q impairments at −f. In addition, from the“Adding Constraints” section (i.e., Case 6, where A and C are constants,and E_(B) and E_(D) are zero) the structure of the single-pointcorrection is known. Using this information, the true single-pointcalibration coefficients α and β can be determined.

The goal is to determine the values α and β given the values A(f),E_(B)(f), C(f) and E_(D)(f). A(f), E_(B)(f), C(f) and E_(D)(f) are knownfrom equations (1.56) through (1.59) given the double-point I/Qimpairments, i.e., the gain imbalance values g₁(f)=g(f) and g₂(f)=g(−f)and phase skew values φ₁(f)=φ(f) and φ₂(f)=φ(−f). The values α and β canbe determined from Γ and θ as shown in the following expressions:

α=Γ sin(θ)  (1.75)

β=Γ cos(θ).  (1.76)

Using the phasor diagram of FIG. 54, sum along the x-axis to obtainequation (1.77), and sum along the y-axis to obtain equation (1.78):

CΓ sin(θ)−E _(D)Γ cos(θ)=−A  (1.77)

CΓ cos(θ)+E _(D)Γ sin(θ)=1−E _(B)  (1.78)

We have relied on the facts that:

HT{sin(t)}=−cos(t),

HT{cos(t)}=sin(t),

where HT denotes the Hilbert Transform. Equations (1.77) and (1.78)imply:

$\begin{matrix}{{\begin{bmatrix}{- E_{D}} & C \\C & E_{D}\end{bmatrix}\begin{bmatrix}{\Gamma \; {\cos (\theta)}} \\{\Gamma \; {\sin (\theta)}}\end{bmatrix}} = \begin{bmatrix}{- A} \\{1 - E_{B}}\end{bmatrix}} & (1.79) \\{\begin{bmatrix}{\Gamma \; {\cos (\theta)}} \\{\Gamma \; {\sin (\theta)}}\end{bmatrix} = {{\frac{1}{C^{2} + E_{D}^{2}}\begin{bmatrix}{- E_{D}} & C \\C & E_{D}\end{bmatrix}}\begin{bmatrix}{- A} \\{1 - E_{B}}\end{bmatrix}}} & (1.80) \\{\alpha = {{\Gamma \; {\sin (\theta)}} = \frac{{- {AC}} + {E_{D}\left( {1 - E_{B}} \right)}}{C^{2} + E_{D}^{2}}}} & (1.81) \\{\beta = {{\Gamma \; {\cos (\theta)}} = \frac{{AE}_{D} + {C\left( {1 - E_{B}} \right)}}{C^{2} + E_{D}^{2}}}} & (1.82) \\{\Gamma = \sqrt{\frac{A^{2} + \left( {1 - E_{B}} \right)^{2}}{C^{2} + E_{D}^{2}}}} & (1.83) \\{\theta = {{\arctan \left( \frac{{- {AC}} + {E_{D}\left( {1 - E_{B}} \right)}}{{AE}_{D} + {C\left( {1 - E_{B}} \right)}} \right)}.}} & (1.84)\end{matrix}$

While not needed to solve for α and β, solving for Γ and θ tell us thenew gain and phase of the waveform needed to exactly cancel the impactsof the I/Q corruption.

It should be noted that the correction coefficients α and β given by(1.81) and (1.82) are generally different from the α and β used intraditional single-point compensation as described in the section“Performing Traditional Impairment Compensation at a Single Frequency”.(Thus, the traditional single-point compensation values will generallygive less than perfect compensation when used as a pre-compensation,i.e., when used in FIGS. 53A and 53B.) However, there is a special casein which the two coefficient pairs coincide. As explained in the“Corruption I/Q Impairments” section, when the gain imbalance and phaseskew functions are even functions, the corruption model values reduceto:

A(f)=g(f)sin(φ(f))

E _(B)(f)=0

C(f)=g(f)cos(φ(f))

E _(D)(f)=0.

Thus, equations (1.81) and (1.82) will specialize to:

α=−tan(φ(f))

β=1/{g(f)cos(φ(f))},

which are the same values used by the traditional single-pointcompensation.

Iterative Technique for Measuring TX Impairments

With reference now to FIG. 55A, the problems of measuring the amplituderesponse of receive filter 5525 and the I/Q impairments of the receiveris simplified (relative to the corresponding problems for thetransmitter) because the I/Q impairments that result from the I/Qdemodulator 5530 occur after the distortion effects of the receivefilter 5525. For example, if a pure tone is the input signal to thereceive path, then the distortion of the receive filter will alter onlythe magnitude and phase of the single tone. Then this altered pure tonewill be distorted by the I/Q demodulator, creating the I/Q impairments.When calibrating the receiver, we can first remove the receiver's I/Qimpairments, leaving only the filter's amplitude and phase responseeffects and then correct for the filter's amplitude and phase distortionin an additional step if desired.

However, this is not the case for the transmitter. Shown in FIG. 55B areis the signal path for the transmitter and receiver in combination. Thetransmitter includes an I/Q modulator 5510 and a transmit filter 5515.In some embodiments, both the transmit and receive LOs are shared. Whenthe transmitter creates a single tone, the I/Q modulator 5510 introducestransmit I/Q impairments. Then these impairments travel through thetransmit signal path, cabling, and the receive signal path beforefinally reaching the I/Q demodulator. This path between the I/Qmodulator output and the I/Q demodulator input corrupt the transmit I/Qimpairments measurement taken at the receiver. In addition, the I/Qimpairments of the demodulator further corrupt the measurement of thetransmitter I/Q impairments taken at the receiver. In alternativeembodiments, the receiver may be based on an alternative RF architecture(i.e., other than a direct conversion architecture) such that the I/Qimpairments of the receiver are very small, i.e., small enough toneglect.

Shown in FIG. 55C is an example of how the non-flat amplitude responsein the signal path corrupts the I/Q impairments seen at the receiver.Coming out of the I/Q modulator are the actual I/Q impairments. Then thetransmit signal path corrupts them, followed by a phase rotation due tothe electrical delay of the cable, followed by another corruption by thereceive signal path. In addition to the amplitude, the phase response(not shown in FIG. 55C) also creates a different but related problem.

Upon initial observation, it would appear that the ideal solution wouldfirst characterize the magnitude and phase of the signal path betweenthe I/Q modulator and the I/Q demodulator. Then the effects of thesignal path could be removed from the receiver-measured I/Q impairmentsby using the calculation in the section “Altering Gain Imbalance andPhase Skew through a Filter”. However, this is not a reasonable taskgiven the performance requirements for the impairment suppression. Inorder to achieve image suppression of better than −80 dB, the phase skewwould need to be less than 0.01 degrees. Even at lower RF frequencies,this means that the absolute phase must be stable and measureable tobetter than picosecond accuracy. In addition, the I/Q impairments alterthe magnitude and phase of the signal coming out of the modulator asdescribed in the section “Magnitude and Phase Corruption from I/QImpairments” and expressed in equation (4.9) of FIG. 58A. As a result,the I/Q impairments of the transmitter, the very thing we are trying tomeasure, would need to be known in order to determine the magnitude andphase response of the signal path.

A better approach to determining the exact I/Q impairments through asignal path is to iterate on the solution. Given a coarse estimate ofthe amplitude and phase of the signal path and an estimate of the I/Qimpairments, the exact I/Q impairments can be determined through enoughiterations. (The iterations may be performed using shared LOs or offsetLOs, as described in detail below. In the case of shared LOs, thereceiver's I/Q impairments need to be known. In the case of offset LOs,the receiver I/Q impairments do not need to be known, although knowingthem helps. In both cases, the transmitter's I/Q impairments do not needto be known beforehand. They will be determined as a result of theiterations.) The total number of iterations will depend largely on theinitial estimates and the performance criteria. Listed below is aprocedure for determining the transmitter's impairments for both sharedand offset LOs. This procedure measures all calibration frequencylocations within the instantaneous bandwidth and only iterates overthese measurements once all measurements for a given instantaneousbandwidth have been completed. Given in the section on Optimizations isa modified procedure that obtains the same result but generally requiresfewer iterations.

Iterative Method Steps (Overview):

1. Tune the RX and TX LOs.

2. Measure RX Impairments.

3. Measure the mapping between RX and TX.

4. Apply estimated impairment correction at TX.

5. Generate tone at TX and measure at RX.

6. Remove RX impairments from #5.

7. Remove signal path estimate (e.g., apply the mapping from #3).

8. Combine results from all iterations of #7 to create an updatedimpairment estimate.

9. If performance metric is acceptable go to #10; else iterate by goingto #4.

10. Repeat steps #1 through #9 for each LO frequency.

Iterative Method Steps (Descriptive)

1. Tune the transmit and receive LOs to the first desired LO frequency.If using shared LOs (using the same LO or using two separate LOs thatare locked together), the LOs will be at the same frequency. In the caseof offset LOs, the LOs are offset from each other by some known exactamount. In either case, ensure that all LOs are phase locked. See the“Restrictions” subsection of the section “Offset LO Method CalibrationMethod” for more information on selecting a working offset. Also keep inmind the window used in the measurement. If using no window, as is donein the “Rectangle Window Optimization” section, be sure that offset LOvalue is confined to frequencies given in that section.

2. (Optional when using the offset LO method) Measure the gain imbalanceand phase skew of the receiver for each in-band offset frequency atwhich the transmitter is to be measured. This can be accomplished byusing the measurement method prescribed in section “Precise MeasurementTechnique”. Since using offset LOs makes the images appear at differentfrequencies for receive and transmit, removing the receive impairmentsis not critical as it is in the case of shared LOs. In all known datasets, this iteration method converges without knowing the receiveimpairments when the LOs are offset. However, the receive impairments docause some corruption to the transmit impairments. As a result, if theyare too severe they would cause this iteration method to diverge ratherthan converge even when using offset LOs.

3. Connect the transmitter's output to the receiver's input.

4. (Only for the offset LO method) Frequency shift the receiver'sspectrum by an amount equal to the LO offset. For example, if thetransmitter's LO is located at 2.400 GHz and the receiver's LO at 2.404GHz, then shift the spectrum by a positive 4 MHz. The frequency shiftmust be frequency locked to the LOs or else the rotation estimate madein step 5 will not remain fixed.

5. Determine the rotation and scaling mapping between receive andtransmit by using the algorithm in the section “Calculating the MappingBetween and RX and TX”. For the best results, apply a tone somewhere inthe instantaneous bandwidth as the leakage can be sensitive to in-bandpower. This mapping should be consistent and repeatable once the LOs aresetup. Thus, in at least some embodiments, the LOs are required to bephase locked. The exact LO offset is known when using the offset LOmethod.

6. If this is the first iteration of #6, do not apply any correction(pass-through) at the transmitter and proceed to #7. Otherwise, apply acorrection filter at the transmitter based upon the measurements in #10.

7. Apply a complex exponential tone at the transmitter for each of thedesired in-band measurement locations, and determine the raw gainimbalances and phase skews by using the calculation method in the“Precise Measurement Technique” section at each of the frequencyoffsets.

8. (Optional when using the offset LO method) For each of the measuredvalues in #7, mathematically remove the receiver's gain imbalance andphase skew. This can be done by the calculation described in section“Removing Receiver Impairments from Measured Output Impairments”. Thisplaces the measurement of the transmitter before the demodulator. Inlieu of step #8, another method is to apply a correction filter at thereceiver before step #7 by computing the needed correction (from thesection “Wideband I/Q Impairment Equalization”) and passing the capturedwaveform through the correction. This method is not as accurate becausethe correction filter has the potential to not be as accurate as themeasurement due to limited filter taps.

9. For each of the calculated values in #8, remove the approximatelyknown rotation, scaling, magnitude, and phase by using the transformdescribed in “Altering the Gain Imbalance and Phase Skew through aLinear System”. The rotation and scaling were determined in step #5.After the first iteration, an estimation of the magnitude can also bedetermined. This places the measurement approximately at the output ofthe modulator. If the measurement were exactly at the output of themodulator, we would not need this iteration approach. This iterativemethod is needed because we do not know within the accuracy needed therotation, scaling, magnitude and phase of the path between the output ofthe modulator and the input of the demodulator.

10. Combine the results from all iterations of #9 by finding the productof all of the gain imbalances (when using the linear scale) and the sumof all of the phase skews on a per frequency offset and LO combinationbasis. For example, if measurements were performed at −15 MHz, −5 MHz, 5MHz, and 15 MHz, then only the measurements taken at −15 MHz arecombined together from other iterations. When moving on to another LO in#13, this combination starts over so that the measurements at −15 MHzand LO=2.4 GHz are not combined with measurements at −15 MHz and LO=2.6GHz)

11. Calculate the image rejection from the gain imbalance and phase skewat each in-band frequency location measured in #9 and calculated byequation 4.15. Determine the worse case image rejection across the bandby finding the minimum of all of the image rejection calculations.

12. If the image rejection from #11 meets the required performancemetrics, the final gain imbalance and phase skew measurements were thosecomputed in step #10 and no more iterations are required for this LOfrequency, otherwise iterate on the solution by going to #6.

13. Repeat steps #1 through #11 for each LO frequency.

In one set of embodiments, the transmitter's I/Q impairments may beestimated according to the method given in Appendix A.

Results

FIGS. 56A and 56B show the improvement (i.e., the rate of convergence)of each iteration according to one embodiment of the iterative method.In at least some embodiments, the iterative method has a convergenceinterval of [−3 dB, 3 dB] for magnitude and a convergence interval of[−30 degrees, 30 degrees] for phase. In these embodiments, if themagnitude or phase have an error outside of these intervals, thesequence of measurements will diverge. FIGS. 56A and 56B show theconvergence per iteration for both a magnitude error and phase error.

Optimizations

This section describes how to optimize the iterative process describedabove to use less total acquisitions and thus less calibration time. Theproblem with the iteration process described above is that it takesmultiple acquisitions for a single wideband measurement within theinstantaneous bandwidth in addition to calculating new filters betweeniterations. However, the total number of acquisitions can be greatlydecreased by using a single point vector calibration to iterate on asingle point to determine its actual impairment value. Then by steppingthrough the band, the previous measurement location of the impairmentsbecomes an estimate for the next measurement location. This works wellwhen the impairments are not changing quickly across the band, thusproviding good estimates for actual values nearby.

By adding this optimization, it is advisable to create a frequency planthat is as follows:

[Δf/2,−Δf/2,2*Δf/2,−2*Δf/2,3*Δf/2,−3*Δf/2, . . . ,N*Δf/2,−N*Δf/2]

for integer N, where Δf is the spacing of frequency measurementlocations within instantaneous bandwidth. This allows for the maximumbenefit of the optimization since it creates the best estimate of thenew point to be measured by using its neighbor. Since this method uses atrue single-point calibration of the transmitter, it requiresinformation about the impairments at both the tone location and itsimage. This is the reason for alternating between positive and negativefrequencies. This alternating frequency plan is also assumed for thefollowing numbered procedure.

Iterative Method Steps Optimized (Descriptive):

1. Tune the transmit and receive LOs to the first desired LO frequency.If using shared LOs (using the same LO or using two separate LOs thatare locked together), the LOs will be at the same frequency. In the caseof offset LOs, the LOs are offset from each other by some known exactamount. In either case, ensure that all LOs are phase locked. See theRestrictions section of Offset LO Method Calibration Method for moreinformation on picking a working offset. Also keep in mind the windowused in the measurement. If using no window, as is done in the RectangleWindow Optimization section, be sure that offset LO value is confined tofrequencies given in that section.

2. (Optional when using the offset LO method) Measure the gain imbalanceand phase skew of the receiver for each in-band offset frequency wherethe transmitter is to be measured. This can be accomplished by using themeasurement method prescribed in the section “Precise MeasurementTechnique”. Since using offset LOs makes the images appear at differentfrequencies for receive and transmit, removing the receive impairmentsis not critical as it is in the case of shared LOs. In all known datasets, this iteration method converges without knowing the receiveimpairments when the LOs are offset. However, the receive impairments docause some corruption to the transmit impairments. As a result, if theyare too severe they would cause this iteration method to diverge ratherthan converge even when using offset LOs.

3. Connect the transmitter's output and receiver's input.

4. (Only for the offset LO method) Frequency shift the receiver'sspectrum by an amount equal to the LO offset. For example, if the TX'sLO is located at 2.4 GHz and the RX's LO at 2.404 GHz, then shift thespectrum by a positive 4 MHz. The frequency shift is frequency locked tothe LOs. (Otherwise, the rotation estimate made in step 5 will notremain fixed.)

5. Determine the rotation and scale mapping between receive and transmitby using the algorithm in “Calculating the Mapping Between and RX andTX”. For the best results, apply a tone somewhere in the instantaneousbandwidth as the leakage can be sensitive to in-band power. This mappingshould remain consistent and repeatable once the LOs are setup. Thus, inat least some embodiments, the LOs are phase locked. The exact LO offsetis known when using the offset LO method.

6. If this is the first iteration of #6 for this particular LOfrequency, do not apply any correction at the transmitter (simply passthrough) and proceed to #7. Optionally, if this is the first iterationof #6 for this particular LO frequency, apply a tone near 0 Hz in step#5 and use the gain imbalance and phase skew information that wasacquired simultaneously with the leakage (0 Hz) information used in thealgorithm to create the initial estimate of the impairments for both thetone and image. Otherwise, apply a single point correction at thetransmitter based upon the measurements below (assuming the frequencyplan provided above) using the calculation found in “Calculating TrueSingle Point Vector Calibration Constants”.

a. If this is the first iteration of #6 since #13, the best toneestimate is found in the variable $Previous_Impairments2. Otherwise, thecurrent value of #10 is the best estimate.

b. The best image estimate is found in the variable$Previous_Impairments1.

7. Apply a complex exponential tone at the transmitter for currentmeasurement location, and determine the raw gain imbalance and phaseskew by using the calculation method in the section “Precise MeasurementTechnique” for this particular in-band frequency offset.

8. (Optional when using the offset LO method) For each of the measuredvalues in #7, mathematically remove the receiver's gain imbalance andphase skew. This can be accomplished by the calculation described in thesection “Removing Receiver Impairments from Measured OutputImpairments”. This places the measurement of the transmitter before thedemodulator. In lieu of step #8, another method is to apply a correctionfilter at the receiver before step #7 by computing the needed correction(from section “Wideband I/Q Impairment Equalization”) and passing thecaptured waveform through the correction. This method is not as accuratebecause the correction filter has the potential to not be as accurate asthe measurement due to limited filter taps.

9. Remove the approximately known rotation, scaling, magnitude, andphase from #8 by using the transform described in “Altering the GainImbalance and Phase Skew through a Linear System”. The rotation andscaling were determined in step #5. A good estimation of the magnitudecan be found by using its neighbor's magnitude in the same way that agood estimation of the impairments are found in step #6. This places themeasurement approximately at the output of the modulator. If themeasurement were exactly at the output of the modulator, we would notneed this iteration approach. This iterative method is needed because wedo not know within the accuracy needed the rotation, scaling, magnitudeand phase of the path between the output of the modulator and the inputof the demodulator.

10. Combine the results from all iterations of #9 and the variable$Previous_Impairments2 by finding the product of all of the gainimbalances (when using the linear scale) and the sum of all of the phaseskews on a per frequency offset and LO combination basis. For example,if measurements were performed at −15 MHz, −5 MHz, 5 MHz, and 15 MHz,then only the measurements taken at −15 MHz are combined together fromother iterations. When moving on to another LO in #13, this combinationstarts over so that the measurements at −15 MHz and LO=2.4 GHz are notcombined with measurements at −15 MHz and LO=2.6 GHz)

11. Calculate the image rejection by using the gain imbalance and phaseskew information from #9 and equation 4.15.

12. If the image rejection from #11 meets the required performancemetrics, the final gain imbalance and phase skew measurements for thecurrent measurement location are those computed in step #10 and no moreiterations are required for this in-band frequency and LO frequencycombination. As a result, advance to #13 and save the value in thevariable $Previous_Impairments1 into $Previous_Impairments2, and storethe current measurements in a variable, $Previous_Impairments1.Otherwise iterate on the solution by going to #6.

13. Repeat steps #6 through #12 for each in-band frequency measurementlocation.

14. Repeat steps #1 through #13 for each LO frequency and clear allvariables.

In some embodiments, the transmitter's I/Q impairments may be estimatedusing offset LOs as described in Appendix B.

In other embodiments, the transmitter's I/Q impairments may be estimatedusing shared LOs as described in Appendix C.

Magnitude and Phase Corruption from I/Q Impairments

This section derives various equations useful for understanding how I/Qimpairments corrupt the magnitude and phase of a signal. We will seethat a signal s(f,t) of the form

s(f,t)=cos(2πft)+jg(f)sin(2πft+φ(f))

includes a tone at frequency f and an image at frequency −f FIG. 57provides notation for the amplitudes of the tone and image. Thederivation, including equations (4.8) through (4.21), is given in FIGS.58A and 58B. Equation (4.11) specifies the amplitude |α| of the tone asthe result of the I/Q impairments. Notice that if the gain imbalance isequal to one and the phase skew is equal to zero, then there is nochange in the tone's amplitude. Additionally, once the impairments areknown, the image rejection can be directly calculated by using equation(4.15).

Precise Measurement Technique

This section describes a method for measuring magnitude, phase, leakage,gain imbalance, and phase skew accurately and quickly. In addition tomeasurement quality and speed, the method also lends itself well to anFPGA implementation for an even greater computational speedup.

This method is a stimulus/response method that injects a known signal atthe input and then measures the output. Specifically, the stimulus is apure complex exponential with a frequency equal to the frequencylocation for the desired measurements. In some embodiments, this complexexponential is generated by a calibration synthesizer or by thetransmitter looped back into the receiver. For each frequency of thecomplex exponential, the response is digitized and processed in order todetermine corresponding measurements. The rest of this section discusseshow the digitized response data is processed to give the measurements ofinterest.

When this processing is thought of in the time domain, the basic idea isto mix each of the signals to DC and then employ averaging to get aprecise result. In the frequency domain, this can be thought of as thecalculation of a few single-point windowed Discrete-Time FourierTransforms. This explanation and derivation will assume the use of arectangle window (with a width equal to the acquisition length) beforecomputing the DTFT. Windowing and its effects are discussed in greatdetail in the next section, “Rectangle Window Optimization”.

Equation 6.1 describes the expected form of the analog response. Thisform assumes the stimulus is a complex exponential at a known frequencyf. Equation 6.3 defines a DTFT that has infinite support, and as aresult, is not realizable for actual computations. Equation 6.4 givesthe DTFT with finite support by using a rectangle window. The value wrepresents the normalized digitized frequency on the interval [−π,π].The conversion from f to w is given by: w=2πf/SampleRate.

Measuring the leakage of the signal does not require shifting andrequires just averaging since its spectral components are alreadylocated at 0 Hz. In order to measure the magnitude and phase of a giventone, first mix the complex tone down to 0 Hz by multiplying by acomplex exponential with a frequency equal and opposite to the tonefrequency. Then average the result over the acquisition length. Thisagain is equivalent to taking a single-point DTFT at the frequency ofinterest for the complex input signal.

$\begin{matrix}\begin{matrix}{{s\left( {t,f} \right)} = {{I\left( {t,f} \right)} + {j\; {Q\left( {t,f} \right)}}}} \\{= {{m(f)}{\exp \left( {{j\theta}(f)} \right)}\left\{ {{\cos \left( {2\pi \; {ft}} \right)} + {j\; {g(f)}{\sin \left( {{2\pi \; {ft}} + {\phi (f)}} \right)}}} \right\}}}\end{matrix} & (6.1) \\{{s\lbrack n\rbrack} = {{ADC\_ Sampling}\left( {s\left( {t,f} \right)} \right)}} & (6.2) \\{{S(w)} = {{{DTFT}\left\{ {s\lbrack n\rbrack} \right\}} = {\sum\limits_{n = {- \infty}}^{\infty}\; {{s\lbrack n\rbrack}{\exp \left( {{- j}\; {wn}} \right)}}}}} & (6.3) \\\begin{matrix}{{W(w)} = {{DTFT}\left\{ {{Window}\left( {s\lbrack n\rbrack} \right)} \right\}}} \\{= {\frac{1}{N}{\sum\limits_{n = 0}^{N - 1}\; {{s\lbrack n\rbrack}{\exp \left( {{- j}\; {wn}} \right)}}}}}\end{matrix} & (6.4) \\\begin{matrix}{{Leakage} = {W(0)}} \\{= {{Average}\left\{ {s\lbrack n\rbrack} \right\}}} \\{= {\frac{1}{N}{\sum\limits_{n = 0}^{N - 1}\; {s\lbrack n\rbrack}}}}\end{matrix} & (6.5) \\\begin{matrix}{{{Magnitude}\left\{ {s\lbrack w\rbrack} \right\}} = {{W(w)}}} \\{= {{\frac{1}{N}{\sum\limits_{n = 0}^{N - 1}\; {{s\lbrack n\rbrack}{\exp \left( {{- j}\; {wn}} \right)}}}}}}\end{matrix} & (6.6) \\{A_{I} = {{Re}\left( {{Avg}\left\{ {{s\lbrack n\rbrack}{\exp \left( {{- j}\; {wn}} \right)}} \right\}} \right)}} & (6.7) \\{A_{Q} = {{Im}\left( {{Avg}\left\{ {{s\lbrack n\rbrack}{\exp \left( {{- j}\; {wn}} \right)}} \right\}} \right)}} & (6.8) \\{{{Magnitude}\left\{ {s\lbrack w\rbrack} \right\}} = \sqrt{A_{I}^{2} + A_{Q}^{2}}} & (6.9) \\{{{Phase}\left\{ {s\lbrack w\rbrack} \right\}} = {{\tan^{- 1}\left( \frac{A_{Q}}{A_{I}} \right)}.}} & (6.10)\end{matrix}$

Alternatively, the phase of {s[n]} may be computed according to theexpression:

$\begin{matrix}{{{Phase}\left\{ {s\lbrack w\rbrack} \right\}} = {\tan^{- 1}{\frac{{Im}\left\{ {{Sum}\left( {{s\lbrack n\rbrack}{\exp \left( {{- j}\; {wn}} \right)}} \right)} \right\}}{{Re}\left\{ {{Sum}\left( {{s\lbrack n\rbrack}{\exp \left( {{- j}\; {wn}} \right)}} \right)} \right\}}.}}} & \left( {6.7\; B} \right)\end{matrix}$

Computing the gain imbalance and phase skew involves finding themagnitude and phase of the I and Q signals independently. For example,in FIG. 59, the “Q actual” signal is a 26 MHz signal with a 0.6 gainimbalance and 20 degree phase skew compared to the in-phase signal(i.e., the “I Reference” signal). However, the “Q Desired” trace givesthe ideal quadrature signal, which is 90 degrees offset from thein-phase signal. By measuring the magnitude and phase of the in-phasecomponent (“I Reference”), the ideal quadrature signal can be determinedby its orthogonality relative to the in-phase component. Then by knowingthe actual magnitude and phase of the quadrature signal (“Q Actual”),the difference between ideal quadrature signal and the actual quadraturesignal can be determined.

Shown in FIGS. 60 and 61 are the magnitudes for the in-phase andquadrature phase signal components, i.e., for the “I Reference” signaland “Q actual signal” in FIG. 59. Since each component of the complexsignal s(t) is a real-valued signal, it is expected to have a symmetricmagnitude response. In order to find the gain imbalance g(f), determinethe gain of each signal component at the frequency location of the tone,and then divide the Q gain by the I gain as given in equation 6.12.

Equations 6.8 through 6.11 show how to compute the magnitude and phaseof each component. Following the convention of assuming the in-phasesignal is perfect and the quadrature phase signal contains all of theimpairments, the impairments are computed relative to the in-phasesignal reference. (Other conventions are possible, as describedvariously above. For example, the quadrature signal could just as wellhave been chosen as the reference.) As a result, the magnitude and phaseare computed for each of the I signal and the Q signal by finding thesingle point DTFT. Then these magnitudes and phases are combinedtogether by equations 6.12 and 6.13 in order to determine the gainimbalance and phase skew of the quadrature signal component.

In the equations below, I(n,w) is the sampled version of I(t,w), andQ(n,w) is the sampled version of Q(t,w).

$\begin{matrix}{{{I(w)}} = {{{Avg}\left\{ {{I\left( {n,w} \right)}{\exp \left( {{- j}\; {wn}} \right)}} \right\}}}} & (6.8) \\{{{Phase}\left\{ {I(w)} \right\}} = {\tan^{- 1}\left( \frac{{Im}\left\{ {{Avg}\left( {{I\left( {n,w} \right)}{\exp \left( {{- j}\; {wn}} \right)}} \right)} \right\}}{{Re}\left\{ {{Avg}\left( {{I\left( {n,w} \right)}{\exp \left( {{- j}\; {wn}} \right)}} \right)} \right\}} \right)}} & (6.9) \\{{{Q(w)}} = {{{Avg}\left\{ {{Q\left( {n,w} \right)}{\exp \left( {{- j}\; {wn}} \right)}} \right\}}}} & (6.10) \\{{{Phase}\left\{ {Q(w)} \right\}} = {\tan^{- 1}\left( \frac{{Im}\left\{ {{Avg}\left( {{Q\left( {n,w} \right)}{\exp \left( {{- j}\; {wn}} \right)}} \right)} \right\}}{{Re}\left\{ {{Avg}\left( {{Q\left( {n,w} \right)}{\exp \left( {{- j}\; {wn}} \right)}} \right)} \right\}} \right)}} & (6.11) \\{{g(w)} = \frac{{Q(w)}}{{I(w)}}} & (6.12) \\{{\phi (w)} = {{{Phase}\left\{ {Q(w)} \right\}} - {{Phase}\left\{ {I(w)} \right\}} + {\pi/2.}}} & (6.13)\end{matrix}$

In an alternative embodiment, ∥I(w)∥, Phase{I(w)}, ∥Q(w)∥ andPhase{Q(w)} may be calculated as follows:

$\begin{matrix}{{{I(w)}} = {{{{Sum}\left\{ {{I\left( {n,w} \right)}{\exp \left( {{- j}\; {wn}} \right)}} \right\}}}/N}} & (6.8) \\{{{Phase}\left\{ {I(w)} \right\}} = {\tan^{- 1}\left( \frac{{Im}\left\{ {{Sum}\left( {{I\left( {n,w} \right)}{\exp \left( {{- j}\; {wn}} \right)}} \right)} \right\}}{{Re}\left\{ {{Sum}\left( {{I\left( {n,w} \right)}{\exp \left( {{- j}\; {wn}} \right)}} \right)} \right\}} \right)}} & (6.9) \\{{{Q(w)}} = {{{{Sum}\left\{ {{Q\left( {n,w} \right)}{\exp \left( {{- j}\; {wn}} \right)}} \right\}}}/N}} & (6.10) \\{{{{Phase}\left\{ {Q(w)} \right\}} = {\tan^{- 1}\left( \frac{{Im}\left\{ {{Sum}\left( {{Q\left( {n,w} \right)}{\exp \left( {{- j}\; {wn}} \right)}} \right)} \right\}}{{Re}\left\{ {{Sum}\left( {{Q\left( {n,w} \right)}{\exp \left( {{- j}\; {wn}} \right)}} \right)} \right\}} \right)}},} & (6.11)\end{matrix}$

where N is the acquisition size.

FIG. 62 illustrates a software embodiment (written in the LabVIEWgraphical programming language) for calculating LO Leakage, Amplitude,Gain Imbalance, Image Rejection and Phase Skew.

In some embodiments, the computations of

Sum{ Re(Q(n,w)exp(−jwn)) } Sum{ Im(Q(n,w)exp(−jwn)) } Sum{ Re(s[n]) }Sum{ Im(s[n]) }are performed by a programmable hardware element (e.g., an FPGA of thereceiver). FIG. 63 shows a LabVIEW graphical program (VI) that receivesthe summation values computed by the FPGA, and computes LO Leakage,amplitude, gain imbalance and phase skew based on those summation valuesand the acquisition length. (Any of the various computer systemsdescribed herein may include software infrastructure for executingcomputer programs including, e.g., LabVIEW graphical programs.)

Rectangle Window Optimization

In some embodiments, a non-rectangular window may be applied to thecomplex digital signal {s[n]}. Any of various standard window types maybe used. In other embodiments, no window is explicitly applied to thecomplex digital signal. However, by performing computations only over afinite acquisition interval, a rectangle window is implicitly beingapplied. If we place frequency-planning restrictions on the placement oftones in the spectrum or judge the calculated measurement error to beacceptable, no window needs to be explicitly applied to the complexdigital signal. (Thus, we can avoid the memory required for storing thewindow values, minimizing the hardware utilization.) Otherwise, a windowshould be used to make the measurements. This section will discuss thederivation for the frequency plan restrictions and the measurement errorthat results if the restricted frequency plan is not used when using nowindow.

The following is the derivation for a rectangular window (i.e., noexplicit window). Just for reference, equation 5.9 is the equation forthe standard DTFT and equation 5.12 gives the closed form solution for afinite geometric series. A rectangle window is defined as unity over afinite interval and zero elsewhere. As a result, its DTFT is given by5.11. Using the geometric identity of equation 5.12, the DTFT of thewindow can be simplified to equation 5.13. Finally, since the first termof the final expression of 5.13 has unit magnitude, the log magnitude isgiven by equation 5.14.

$\begin{matrix}\begin{matrix}{{X(w)} = {{DTFT}\left\{ {x\lbrack n\rbrack} \right\}}} \\{= {\sum\limits_{n = {- \infty}}^{\infty}\; {{x\lbrack n\rbrack}{\exp \left( {{- j}\; {wn}} \right)}}}}\end{matrix} & (5.9) \\{{{rect}\lbrack n\rbrack} = \left\{ \begin{matrix}1 & {{{if}\mspace{14mu} 0} \leq n \leq {M - 1}} \\0 & {otherwise}\end{matrix} \right.} & (5.10) \\{{{RECT}(w)} = {\sum\limits_{n = 0}^{M - 1}\; {\exp \left( {{- j}\; {wn}} \right)}}} & (5.11) \\{{\sum\limits_{n = 0}^{M - 1}\; a^{n}} = {{\frac{1 - a^{M}}{1 - a}\mspace{14mu} {for}\mspace{14mu} a} \neq 1}} & (5.12) \\\begin{matrix}{{{RECT}(w)} = {\sum\limits_{n = 0}^{M - 1}\; {\exp \left( {{- j}\; {wn}} \right)}}} \\{= \frac{1 - {\exp \left( {{- j}\; {wM}} \right)}}{1 - {\exp \left( {{- j}\; w} \right)}}} \\{= {\left\lbrack \frac{\exp \left( {{- j}\; {{wM}/2}} \right)}{\exp \left( {{- j}\; {w/2}} \right)} \right\rbrack \left\lbrack \frac{\sin \left( {{wM}/2} \right)}{\sin \left( {w/2} \right)} \right\rbrack}}\end{matrix} & (5.13) \\{{{{RECT}(w)}} = {20\; {\log \left( \frac{\sin \left( {{wM}/2} \right)}{\sin \left( {w/2} \right)} \right)}}} & (5.14)\end{matrix}$

Note that for a pure tone, the nulls in the spectrum of the windowedtone will occur at

Ftone+/−N*SampleRate/AcqLength,

Ftone is the tone frequency, and AcqLength is the number of samples inthe acquisition of the complex digital signal, and SampleRate is therate at which the samples of the complex digital signal are acquired.Also note that for the image rejection calculation, if we make sure thatall generated tones exist only at multiples of SampleRate/AcqLength,then there won't be any spectral leakage effects in the measurement.

FIGS. 64-65 show two respective plots of the amplitude spectrum|RECT(w)| with a common sample rate of 120 MHz and with differentacquisition lengths. The first plot (FIG. 64) corresponds to anacquisition length of 20. The second plot (FIG. 65) corresponds to anacquisition length of 128.

Generalized Derivation

Given the system model of FIG. 66, we can derive the function forms forfrequency responses U(ω) and V(ω) from the input I/Q impairmentsg_(in)(ω) and φ_(in)(ω) and the output I/Q impairments g_(out)(ω) andφ_(out)(ω). Furthermore, we can derive the output impairments from thefrequency responses U(ω) and V(ω) and the input I/Q impairments. Bothderivations rely on the following preliminaries. The system modelimplies:

u(t)*cos(ωt)+v(t)*g _(in)(ω)sin(φt+φ _(in)(ω))=g _(out)(ω)sin(ωt+ω_(out)(ω)),  (7.5)

where u(t) and v(t) are the impulses responses corresponding to U(ω) andV(ω) respectively.

Using standard identities for the cosine and sine functions, we obtain:

U(ω)exp(jωt)/2+U(−ω)exp(−jωt)/2+V(ω)g _(in)(ω)exp{j(ωt+φ_(in)(ω))}/2j−V(−ω)g _(in)(ω)exp{−j(ωt+φ _(in)(ω))}/2j=g_(out)(ω)exp{j(φt+φ _(out)(ω))}/2j−g _(out)(ω)exp{−j(ωt+φ_(out)(ω))}/2j.  (7.6)

Collecting the coefficients of terms in exp(jωt) and separately thecoefficients of the terms in exp(−jωt) gives the following twoequations:

jU(ω)+V(ω)g _(in)(ω)exp(jφ _(in)(ω))=g _(out)(ω)exp(jφ _(out)(ω))  (7.7)

jU(−ω)−V(−ω)g _(in)(ω)exp(−jφ _(in)(ω))=−g _(out)(ω)exp(−jφ_(out)(ω)).  (7.8a)

However, equation (7.8a) applies for all ω. So we can substitute −ω forω, and obtain:

jU(ω)−V(ω)g _(in)(−ω)exp(−jφ _(in)(−ω))=−g _(out)(−ω)exp(jφ_(out)(−ω)).  (7.8b)

Equations (7.7) and (7.8b) define a 2×2 matrix equation in vectorunknown [U(ω),V(ω)]^(T), whose solution is given by equations (7.9) and(7.10) in FIG. 67.

Now, given the input impairments and the frequency response of thefilters U(ω) and V(ω), we derive the output impairments. It may appearfrom equations (7.7) and (7.8a) that calculating the output impairmentsis not possible because the problem is over determined. However, sinceU(ω) and V(ω) are both real-valued filters, there is a directrelationship between their positive and negative frequency responses,i.e., U(−f)=U*(f) and V(−f)=V*(f). Therefore,

g _(out)(ω)=Magnitude{jU(ω)+V(ω)g _(in)(ω)exp(jφ _(in)(ω))}  (7.11)

φ_(out)(ω)=Phase{jU(ω)+V(ω)g _(in)(ω)exp(jφ _(in)(ω))}.  (7.12)

Removing Receiver Impairments from Measured Output Impairments

In this section, we derive a method for computing the input impairmentsg_(in)(f) and φ_(in)(f) of a system given the output impairmentsg_(out)(f) and ω_(out)(f) and the system-intrinsic impairmentsg_(sys)(f) and φ_(sys)(f). This method may be applied to remove thereceiver-intrinsic impairments from the measured impairments at theoutput of the receiver (e.g., the output of the I/Q demodulator) inorder to determine the impairments at the input of the receiver (e.g.,the input of the I/Q demodulator). Given the frequency responses U(f)and V(f) and the output impairments g_(out)(f) and ω_(out)(f) for thesystem model of FIG. 66, we can compute the input impairments g_(in)(f)and ω_(out)(f) starting from equation (7.7), which is copied here interms of frequency f instead of ω:

jU(f)+V(f)g _(in)(f)exp(jφ _(in)(f))=g _(out)(f)exp(jφ_(out)(f))  (7.13)

g _(in)(f)exp(jφ _(in)(f))={−jU(f)+g _(out)(f)exp(jφ_(out)(f))}/V(f).  (7.14)

If we define

Z _(in)(f)=g _(in)(f)exp(jφ _(in)(f)) and  (7.15)

Z _(out)(f)=g _(out)(f)exp(jφ _(out)(f)),  (7.16)

equation (7.14) can be more compactly expressed as:

Z _(in)(f)={−jU(f)+Z _(out)(f)}/V(f).  (7.17)

We can determine U(f) and V(f) from equations (7.9) and (7.10) of FIG.67 by using the special assumptions that g_(in)(f) is identically equalone, φ_(in)(f) is identically equal to zero, g_(out)(f) equals the gainimbalance g_(sys)(f) of the system, and φ_(out)(f) equals the phase skewφ_(sys)(f) of the system. Under these special assumptions, equations(7.9) and (7.10) specialize to:

U(f)=(j/2){g _(sys)(−f)exp(−jφ _(sys)(−f))−g _(sys)(f)exp(jφ_(sys)(f))}  (7.18)

V(f)=(½){g _(sys)(f)exp(jφ _(sys)(f))+g _(sys)(−f)exp(−jφ_(sys)(−f))}.  (7.19)

If we define

Z _(sys)(f)=g _(sys)(f)exp(jφ _(sys)(f)),  (7.20)

equations (7.15) and (7.16) can be expressed as:

U(f)=(j/2){Z _(sys)(−f)*−Z _(sys)(f)}  (7.21)

V(f)=(½){Z _(sys)(f)+Z _(sys)(−f)*}.  (7.22)

By substituting these expressions into equation (7.17), we obtain:

$\begin{matrix}{{Z_{in}(f)} = {\frac{{2\; {Z_{out}(f)}} + {Z_{sys}^{*}\left( {- f} \right)} - {Z_{sys}(f)}}{{Z_{sys}(f)} + {Z_{sys}^{*}\left( {- f} \right)}}.}} & (7.23) \\{{g_{in}(f)} = {{\frac{{2\; {Z_{out}(f)}} + {Z_{sys}^{*}\left( {- f} \right)} - {Z_{sys}(f)}}{{Z_{sys}(f)} + {Z_{sys}^{*}\left( {- f} \right)}}}.}} & (7.24) \\{{\phi_{in}(f)} = {{Phase}{\left\{ \frac{{2\; {Z_{out}(f)}} + {Z_{sys}^{*}\left( {- f} \right)} - {Z_{sys}(f)}}{{Z_{sys}(f)} + {Z_{sys}^{*}\left( {- f} \right)}} \right\}.}}} & (7.25)\end{matrix}$

This calculation method specified by equations (7.23) through (7.25) maybe applied to remove the receiver-intrinsic impairments g_(RX)(f) andφ_(RX)(f) from the measured impairments g_(M)(f) and φ_(M)(f) at theoutput of the receiver (e.g., the output of the I/Q demodulator) inorder to obtain the impairments g_(in)(f) and φ_(in)(f) at the input ofthe receiver (e.g., the input of the I/Q demodulator) as follows:

$\begin{matrix}{{Z_{in}(f)} = {\frac{{2\; {Z_{M}(f)}} + {Z_{RX}^{*}\left( {- f} \right)} - {Z_{RX}(f)}}{{Z_{RX}(f)} + {Z_{RX}^{*}\left( {- f} \right)}}.}} & (7.26) \\{{g_{in}(f)} = {{\frac{{2\; {Z_{M}(f)}} + {Z_{RX}^{*}\left( {- f} \right)} - {Z_{RX}(f)}}{{Z_{RX}(f)} + {Z_{RX}^{*}\left( {- f} \right)}}}.}} & (7.27) \\{{\phi_{in}(f)} = {{Phase}{\left\{ \frac{{2\; {Z_{M}(f)}} + {Z_{RX}^{*}\left( {- f} \right)} - {Z_{RX}(f)}}{{Z_{RX}(f)} + {Z_{RX}^{*}\left( {- f} \right)}} \right\}.}}} & (7.28)\end{matrix}$

Additional embodiments are disclosed in the following numberedparagraphs.

1. A method for operating a receiver, the method comprising:

receiving an analog input signal from a communication medium;

performing I/Q demodulation on the analog input signal to produce ananalog inphase signal and an analog quadrature signal;

digitizing the analog inphase signal and the analog quadrature signal toproduce respectively a digital inphase signal I(n) and a digitalquadrature signal Q(n);

transforming the digital inphase signal I(n) and the digital quadraturesignal Q(n) to produce a resultant digital inphase signal I_(R)(n) and aresultant digital quadrature signal Q_(R)(n) in accordance with theexpressions:

I _(R)(n)=I(n),

Q _(R)(n)=a*I(n)+HT{b*I(n)}+c*Q(n)+HT{d*Q(n)},

where HT denotes the Hilbert Transform, where the coefficients a, b, cand d are computed to achieve at least partial compensation for I/Qimpairments of the receiver at a frequency f and at frequency −f, whereeach of the coefficients is computed based on measured I/Q impairmentsof the receiver at frequency f and measured I/Q impairments of thereceiver at frequency −f.

1B. The method of paragraph 1, where, as an alternative to theexpressions given above, the resultant digital inphase signal IR(n) andthe resultant digital quadrature signal QR(n) are transformed inaccordance with the expressions:

I _(R)(n)=a*I(n)+HT{b*I(n)}+c*Q(n)+HT{d*Q(n)},

Q _(R)(n)=Q(n).

2. The method of paragraph 1, where the analog input signal is a puretone.

3. The method of paragraph 1, where the analog input signal is acommunication signal that carries a stream of binary information.

4. A receiver comprising:

an I/Q demodulator configured to receive an analog input signal, andperform I/Q demodulation on the analog input signal to produce an analoginphase signal and an analog quadrature signal;

a digitization unit configure to digitize the analog inphase signal andthe analog quadrature signal to produce respectively a digital inphasesignal I(n) and a digital quadrature signal Q(n);

a digital circuit configured to transform the digital inphase signalI(n) and the digital quadrature signal Q(n) to produce a resultantdigital inphase signal I_(R)(n) and a resultant digital quadraturesignal Q_(R)(n) in accordance with the expressions:

I _(R)(n)=I(n),

Q _(R)(n)=a*I(n)+HT{b*I(n)}+c*Q(n)+HT{d*Q(n)},

where HT denotes the Hilbert Transform, where the coefficients a, b, cand d are computed to at least partially compensate for I/Q impairmentsof the receiver at a frequency f and at frequency −f, where each of thecoefficients is computed based on measured I/Q impairments of thereceiver at frequency f and measured I/Q impairments of the receiver atfrequency −f.

4B. The receiver of paragraph 4, where, as an alternative to theexpressions given above, the resultant digital inphase signal I_(R)(n)and the resultant digital quadrature signal Q_(R)(n) are transformed inaccordance with the expressions:

I _(R)(n)=a*I(n)+HT{b*I(n)}+c*Q(n)+HT{d*Q(n)},

Q _(R)(n)=Q(n).

5. The receiver of paragraph 4, where the analog input signal is a puretone.

6. The receiver of paragraph 4, where the analog input signal is acommunication signal that carries a stream of binary information.

7. A method for operating a transmitter, the method comprising:

receiving a digital inphase signal I(n) and a digital quadrature signalQ(n);

transforming the digital inphase signal I(n) and the digital quadraturesignal Q(n) to obtain a resultant digital inphase signal I_(R)(n) and aresultant digital quadrature signal Q_(R)(n) in accordance theexpressions:

I _(R)(n)=I(n),

Q _(R)(n)=a*I(n)+HT{b*I(n)}+c*Q(n)+HT{d*Q(n)},

where HT denotes the Hilbert Transform, where the coefficients a, b, cand d are computed to at least partially pre-compensate for I/Qimpairments of the transmitter at frequency f and frequency −f, whereeach of the coefficients is computed based on an estimate of the I/Qimpairments of the transmitter at frequency f and an estimate of the I/Qimpairments of the transmitter at frequency −f.

converting the resultant digital inphase signal I_(R)(n) and theresultant digital quadrature signal QR(n) to analog form in order toobtain respectively an analog I signal and an analog Q signal;

performing I/Q modulation on the analog I and Q signals to produce amodulated analog signal.

7B. The method of paragraph 7, where, as an alternative to theexpressions given above, the resultant digital inphase signal I_(R)(n)and the resultant digital quadrature signal Q_(R)(n) are transformed inaccordance with the expressions:

I _(R)(n)=a*I(n)+HT{b*I(n)}+c*Q(n)+HT{d*Q(n)},

Q _(R)(n)=Q(n).

8. The method of paragraph 7, where the digital inphase signal and thedigital quadrature signal represent a complex exponential tone atfrequency f.

9. The method of paragraph 7, where the digital inphase signal and thedigital quadrature signal carry respective streams of binaryinformation.

10. A transmitter comprising:

a digital circuit configured to receive a digital inphase signal I(n)and a digital quadrature signal Q(n), and transform the digital inphasesignal I(n) and the digital quadrature signal Q(n) to obtain a resultantdigital inphase signal I_(R)(n) and a resultant digital quadraturesignal Q_(R)(n) in accordance with the expressions:

I _(R)(n)=I(n),

Q _(R)(n)=a*I(n)+HT{b*I(n)}+c*Q(n)+HT{d*Q(n)},

where HT denotes the Hilbert Transform, where the coefficients a, b, cand d are computed to at least partially pre-compensate for I/Qimpairments of the transmitter at frequency f and frequency −f, whereeach of the coefficients is computed based on an estimate of the I/Qimpairments of the transmitter at frequency f and an estimate of the I/Qimpairments of the transmitter at frequency −f;

a digital-to-analog conversion (DAC) unit configured to convert theresultant digital inphase signal and the resultant digital quadraturesignal to analog form in order to obtain respectively an analog I signaland an analog Q signal;

an I/Q modulator configured to perform I/Q modulation on the analog Iand Q signals to produce a modulated analog signal.

10B. The transmitter of paragraph 10, where, as an alternative to theexpressions given above, the resultant digital inphase signal I_(R)(n)and the resultant digital quadrature signal Q_(R)(n) are transformed inaccordance with the expressions:

I _(R)(n)=a*I(n)+HT{b*I(n)}+c*Q(n)+HT{d*Q(n)},

Q _(R)(n)=Q(n).

11. The transmitter of paragraph 10, where the digital inphase signaland the digital quadrature signal represent a complex exponential toneat frequency f.

12. The transmitter of paragraph 10, where the digital inphase signaland the digital quadrature signal carry respective streams of binaryinformation.

Yet additional embodiments are disclosed in the following numberedparagraphs.

1. A method for correcting for I/Q impairments in a receivedtransmission signal, the method comprising: receiving the transmissionsignal over a communication medium; performing I/Q demodulation on thereceived transmission signal to produce analog I (in-phase) and Q(quadrature) signals; performing analog to digital conversion of each ofthe analog I and Q signals to produce digital I and Q signals; andperforming wideband I/Q impairment correction on the digital I and Qsignals, where said wideband I/Q impairment correction compensates forfrequency-dependent variations of gain imbalance and phase imbalance inthe digital I and Q signals.

2. The method of paragraph 1, where said wideband I/Q impairmentcorrection compensates for frequency-dependent variations of gainimbalance and phase imbalance in the digital I and Q signals caused byone or more of the I/Q demodulation or the analog to digital conversionof the analog I and Q signals.

3. The method of paragraph 1, where the method is implemented by areceiver device, where said wideband I/Q impairment correctioncompensates for gain imbalances and phase imbalances in the digital Iand Q signals at a plurality of frequency offsets across aninstantaneous bandwidth of the receiver device.

4. The method of paragraph 1, where performing the wideband I/Qimpairment correction on the digital I and Q signals comprises filteringone or more of the digital I signal or the digital Q signal.

5. The method of paragraph 4, where performing the wideband I/Qimpairment correction on the digital I and Q signals comprises filteringthe digital Q signal and leaving the digital I signal unchanged.

6. The method of paragraph 4, where performing the wideband I/Qimpairment correction on the digital I and Q signals comprises filteringthe digital I signal and leaving the digital Q signal unchanged.

7. The method of paragraph 4, where performing the wideband I/Qimpairment correction on the digital I and Q signals comprises filteringboth the digital I signal and the digital Q signal.

8. The method of paragraph 1, where the method is implemented by areceiver device, where the method further comprises determiningcorrection information by providing a plurality of known test signals tothe receiver device and measuring I/Q impairments introduced by thereceiver device in response to the known test signals, where saidwideband I/Q impairment correction utilizes the correction informationto compensate for the frequency-dependent variations of gain imbalanceand phase imbalance in the digital I and Q signals.

9. The method of paragraph 8, where providing the plurality of knowntest signals to the receiver device comprises providing one or more of:a plurality of sine waves at different frequencies; or a plurality ofcosine waves at different frequencies.

10. The method of paragraph 1, where receiving the transmission signalover the communication medium comprises receiving the transmissionsignal over one or more of: a wireless communication medium; or a cable.

11. The method of paragraph 1, where the received transmission signal isa radio frequency (RF) signal.

12. A receiver device configured to: receive a transmission signal overa communication medium; perform I/Q demodulation on the receivedtransmission signal to produce analog I (in-phase) and Q (quadrature)signals; perform analog to digital conversion of each of the analog Iand Q signals to produce digital I and Q signals; and perform widebandI/Q impairment correction on the digital I and Q signals, where saidwideband I/Q impairment correction compensates for frequency-dependentvariations of gain imbalance and phase imbalance in the digital I and Qsignals.

13. The receiver device of paragraph 12, where the receiver deviceincludes: one or more input ports for receiving the transmission signal;one or more output ports for outputting one or more of a correcteddigital I signal or a corrected digital Q signal; and a programmablehardware element configured to perform the wideband I/Q impairmentcorrection.

14. The receiver device of paragraph 13, where the programmable hardwareelement comprises a FPGA (field-programmable gate array).

19. A method for correcting for I/Q impairments, the method comprising:receiving digital I (in-phase) and Q (quadrature) signals to betransmitted; performing wideband I/Q impairment pre-correction on thedigital I and Q signals, where performing the wideband I/Q impairmentpre-correction filters one or more of the digital I and Q signals toproduce one or more pre-corrected digital signals to pre-compensate forfrequency-dependent variations of gain imbalance and phase imbalancethat will be subsequently introduced during synthesis of a transmissionsignal; and synthesizing the transmission signal using the one or morepre-corrected digital signals.

20. The method of paragraph 19, where performing the wideband I/Qimpairment pre-correction filters the digital Q signal to produce apre-corrected digital Q signal and leaves the digital I signalunchanged; where the transmission signal is synthesized from thepre-corrected digital Q signal and the unchanged digital I signal.

21. The method of paragraph 19, where performing the wideband I/Qimpairment pre-correction filters the digital I signal to produce apre-corrected digital I signal and leaves the digital Q signalunchanged; where the transmission signal is synthesized from thepre-corrected digital I signal and the unchanged digital Q signal.

22. The method of paragraph 19, where performing the wideband I/Qimpairment pre-correction filters the digital I signal to produce apre-corrected digital I signal and filters the digital Q signal toproduce a pre-corrected digital Q signal; where the transmission signalis synthesized from the pre-corrected digital I signal and thepre-corrected digital Q signal.

23. The method of paragraph 19, where synthesizing the transmissionsignal comprises: performing digital to analog conversion of the one ormore pre-corrected digital signals to produce one or more of an analog Isignal or an analog Q signal; and performing I/Q modulation to producethe transmission signal using the one or more of the analog I signal orthe analog Q signal; where the one or more pre-corrected digital signalspre-compensate for frequency-dependent variations of gain imbalance andphase imbalance caused by one or more of the digital to analogconversion or the I/Q modulation.

24. The method of paragraph 23, where performing the digital to analogconversion of the one or more pre-corrected digital signals produces ananalog Q signal; where the method further comprises performing digitalto analog conversion of the digital I signal to produce an analog Isignal; where performing the I/Q modulation to produce the transmissionsignal uses the analog Q signal and the analog I signal.

25. The method of paragraph 19, where the method is implemented by atransmitter device; where said wideband I/Q impairment pre-correctionpre-compensates for gain imbalances and phase imbalances at a pluralityof frequency offsets across an instantaneous bandwidth of thetransmitter device.

26. The method of paragraph 19, where the method is implemented by atransmitter device; where the method further comprises determiningcorrection information by providing a plurality of known test signals tothe transmitter device and measuring I/Q impairments introduced by thetransmitter device in response to the known test signals; where saidwideband I/Q impairment pre-correction utilizes the correctioninformation to produce the one or more pre-corrected digital signals.

27. The method of paragraph 26, where providing the plurality of knowntest signals to the transmitter device comprises providing one or moreof: a plurality of sine waves at different frequencies; or a pluralityof cosine waves at different frequencies.

28. The method of paragraph 19, further comprising transmitting thetransmission signal over one or more of: a wireless communicationmedium; or a cable.

29. The method of paragraph 19, where the transmission signal is a radiofrequency (RF) signal.

30. A transmitter device configured to: receive digital I (in-phase) andQ (quadrature) signals to be transmitted; perform wideband I/Qimpairment pre-correction on the digital I and Q signals, whereperforming the wideband I/Q impairment pre-correction filters one ormore of the digital I and Q signals to produce one or more pre-correcteddigital signals to pre-compensate for frequency-dependent variations ofgain imbalance and phase imbalance that will be subsequently introducedduring synthesis of a transmission signal; and synthesize thetransmission signal using the one or more pre-corrected digital signals.

31. The transmitter device of paragraph 30, where the transmitter deviceincludes: one or more input ports for receiving the digital I and Qsignals; one or more output ports for outputting the transmissionsignal; and a programmable hardware element configured to perform thewideband I/Q impairment pre-correction on the digital I and Q signals.

32. The transmitter device of paragraph 31, where the programmablehardware element comprises a FPGA (field-programmable gate array).

34. A measurement system including: a receiver device; and a deviceunder test; where the receiver device is configured to: receive atransmission signal including measurement data acquired from the deviceunder test; perform I/Q demodulation on the received transmission signalto produce analog I (in-phase) and Q (quadrature) signals; performanalog to digital conversion of each of the analog I and Q signals toproduce digital I and Q signals; and perform wideband I/Q impairmentcorrection on the digital I and Q signals, where said wideband I/Qimpairment correction compensates for frequency-dependent variations ofgain imbalance and phase imbalance in the digital I and Q signals.

35. The measurement system of paragraph 34, further comprising: atransmitter device, where the transmitter device is configured to:receive digital I and Q signals to be transmitted, where the digital Iand Q signals specify information to be transmitted to the device undertest; perform wideband I/Q impairment pre-correction on the digital Iand Q signals, where performing the wideband I/Q impairmentpre-correction filters one or more of the digital I and Q signals toproduce one or more pre-corrected digital signals to pre-compensate forfrequency-dependent variations of gain imbalance and phase imbalancethat will be subsequently introduced during synthesis of a transmissionsignal; synthesize the transmission signal using the one or morepre-corrected digital signals; and transmit the transmission signal tothe device under test.

36. The measurement system of paragraph 35, where the transmissionsignal comprises a control signal for controlling the device under test.

37. The measurement system of paragraph 34, further comprising: achassis; where the receiver device is implemented as a first moduleinstalled in the chassis; where the transmitter device is implemented asa second module installed in the chassis.

38. The measurement system of paragraph 37, where the chassis is a PXI(PCI eXtensions for Instrumentation) chassis.

FIG. 68 illustrates one embodiment of a computer system 6800 that may beused to perform any of the method embodiments described herein, or, anycombination of the method embodiments described herein, or any subset ofany of the method embodiments described herein, or, any combination ofsuch subsets.

Computer system 6800 may include a processing unit 6810, a system memory6812, a set 6815 of one or more storage devices, a communication bus6820, a set 6825 of input devices, and a display system 6830.

System memory 6812 may include a set of semiconductor devices such asRAM devices (and perhaps also a set of ROM devices).

Storage devices 6815 may include any of various storage devices such asone or more memory media and/or memory access devices. For example,storage devices 6815 may include devices such as a CD/DVD-ROM drive, ahard disk, a magnetic disk drive, magnetic tape drives, etc.

Processing unit 6810 is configured to read and execute programinstructions, e.g., program instructions stored in system memory 6812and/or on one or more of the storage devices 6815. Processing unit 6810may couple to system memory 6812 through communication bus 6820 (orthrough a system of interconnected busses, or through a network). Theprogram instructions configure the computer system 6800 to implement amethod, e.g., any of the method embodiments described herein, or, anycombination of the method embodiments described herein, or, any subsetof any of the method embodiments described herein, or any combination ofsuch subsets.

Processing unit 6810 may include one or more processors (e.g.,microprocessors).

One or more users may supply input to the computer system 6800 throughthe input devices 6825. Input devices 6825 may include devices such as akeyboard, a mouse, a touch-sensitive pad, a touch-sensitive screen, adrawing pad, a track ball, a light pen, a data glove, eye orientationand/or head orientation sensors, a microphone (or set of microphones),or any combination thereof.

The display system 6830 may include any of a wide variety of displaydevices representing any of a wide variety of display technologies. Forexample, the display system may be a computer monitor, a head-mounteddisplay, a projector system, a volumetric display, or a combinationthereof. In some embodiments, the display system may include a pluralityof display devices. In one embodiment, the display system may include aprinter and/or a plotter.

In some embodiments, the computer system 6800 may include other devices,e.g., devices such as one or more graphics accelerators, one or morespeakers, a sound card, a video camera and a video card, a dataacquisition system.

In some embodiments, computer system 6800 may include one or morecommunication devices 6835, e.g., a network interface card forinterfacing with a computer network. As another example, thecommunication device 6835 may include a specialized interface forcommunication via any of a variety of established communicationstandards or protocols (e.g., USB, Firewire, PCI, PCI Express, PXI).

The computer system may be configured with a software infrastructureincluding an operating system, and perhaps also, one or more graphicsAPIs (such as OpenGL®, Direct3D, Java 3D™). In some embodiments, thesoftware infrastructure may include National Instruments LabVIEW™software, and/or, LabVIEW™ FPGA.

In some embodiments, the computer system 6800 may be configured tointerface with transmitter 6840. The transmitter may be configured totransmit signals (onto a communication channel) as variously describedherein. The transmitter may operate under the control of softwareexecuting on processor 6810 and/or software executing on the transmitteritself.

In some embodiments, the computer system 6800 may be configured tointerface with a receiver 6850. The receiver may be configured toreceive signals (from a communication channel) as variously describedherein. The receiver may operate under the control of software executingon processor 6810 and/or software executing on the receiver itself.

In some embodiments, the transmitter and/or the receiver may include oneor more programmable hardware elements and/or one or moremicroprocessors for performing digital processing on digital data (e.g.,on digital baseband signals or digital IF signals) as variouslydescribed herein.

Although the embodiments above have been described in considerabledetail, numerous variations and modifications will become apparent tothose skilled in the art once the above disclosure is fully appreciated.It is intended that the following claims be interpreted to embrace allsuch variations and modifications.

APPENDIX A Iterative Method for Estimating Transmitter I/Q ImpairmentsUsing Shared LOs 1. Measure the gain imbalance gR and phase skew φR ofthe receiver for each in-band offset frequency f at which thetransmitter's gain imbalance gT and phase skew φT are to be measured.(In some embodiments, the set of frequency offsets is symmetric aboutzero, i.e., for each frequency offset f in the set, the frequency offset−f is also in the set.) For each f, direct a tone generator to generatea tone at frequency v=f_(LO)+f, where f_(LO) is the LO frequency, applythe tone to the receiver's input, and capture the complex basebandsequence z(n) at the output of the receiver's I/Q demodulator. The gainimbalance gR and phase skew φR are computed based on the complexbaseband sequence z(n) as described in the “Precise MeasurementTechnique” section. 2. Configure the receiver and transmitter so thatthey use the same LO frequency f_(LO). If the receiver and transmitteruse two different LO circuits, tune the transmitter so that its LO isphase locked to the same reference. Therefore, the frequency of thetransmitter and the frequency of the receiver are both f_(LO). 3.Connect the transmitter's output to the receiver's input, e.g., via acable or a wireless connection. 4. Estimate the DC scaling m(0) and DCrotation θ(0) of the signal path between transmitter's I/Q modulator andthe receiver's I/Q demodulator by using the algorithm in the section“Calculating the Mapping Between and RX and TX”. For the best results,apply a tone K to the transmitter's I/Q modulator in addition to the DCtest vector. The tone K is applied because the leakage can be sensitiveto in-band power. The tone K is applied at some frequency in theinstantaneous bandwidth other than DC. (As part of the estimation of theDC scaling and DC rotation, the method of the “Precise MeasurementTechnique” section is applied to sampled complex data. If the sampledcomplex data is not windowed, then there are restrictions on thefrequency placement of the tone K.) 5. Iteration index k ← 0 Do while(quality measure Q is smaller than threshold)  For each frequency offsetf:   Set gT(f,0)←1 and φT(f,0)←0.   6A. If k = 0:    Apply nopre-correction at the transmitter, i.e., configure the transmitter'spre-    correction circuitry to use the values α=0 and β=1.   Else (k >0)    Compute pre-correction coefficients α and β for frequency offset fbased on:    current transmitter gain imbalance estimate gT(f,k);current transmitter phase    skew estimate φT(f,k); current transmittergain imbalance estimate gT(−f,k);    current transmitter phase skewestimate φT(−f,k). (If the frequency offset set is    not symmetricabout zero, select the frequency closest to −f for gT(−f,k) and   φT(−f,k).) Alternatively, one could create transmitter pre-correctionfilters.   Endif   Configure the pre-correction circuitry to use thecomputed values α and β (or the   pre-correction filters).   7A.  Applya complex exponential signal u(n)=exp(j2πfn) to the inputs of the pre-  correction circuitry.   7B. Measure complex baseband signal z(n) atthe output of the receiver's I/Q   demodulator.   7C.  Determine rawgain imbalance gz(f) and raw phase skew φz(f) based on the   complexbaseband signal z(n) using the calculation method in “Precise  Measurement Technique”.   8. Remove the receiver's gain imbalancegR(f) and phase skew φR(f) from the raw   gain imbalance gz(f) and rawphase skew φz(f) to obtain pre-demodulation gain   imbalance gPD(f) andpre-demodulation phase skew φPD(f).  (There are at least   two methodsfor performing this removal: a direct-transformation method and a  filtering method. The direct-transformation method may be of higherquality than   the filtering method. The direct-transformation method isdiscussed in the section   entitled “Removing Receiver Impairments fromMeasured Output Impairments”.   The filtering method involves applying a2×2 matrix of digital filters to the   complex baseband signalz(n)=(I(n),Q(n)) to obtain a partially-corrected signal   PCS(n). The2×2 matrix of digital filters may be computed as described above in  connection with Figures 2A, 2B and 3 and in the section “Wideband I/Q  Impairment Equalization”.   9. Remove the best current estimate of thesignal path between the transmitter's I/Q   modulator and the receiver'sI/Q demodulator. m(0) and θ(0) will provide a basic   estimate. Betterestimates will increase the rate of convergence. For example, step   9may be implemented as follows.   If k=0    Remove the estimated DCscaling m(0) and DC rotation θ(0) from the gain    imbalance gPD and thephase skew φPD to obtain post-modulation gain    imbalance gPM andpost-modulation phase skew φPM using the transform    described in“Altering the Gain Imbalance and Phase Skew through a Linear    System”.Set H(f) and H(−f) equal to H(0)=exp(−jθ(0))/m(0).   Else (k>0)   Compute scaling m(f) at frequency offset f based on the complexbaseband    signal z(n). The scaling m(f) may be determined by computingthe magnitude of    the frequency component at frequency f in thecomplex signal z(n), as explained    in the “Precise MeasurementTechnique” section, especially in equation 6.6.    Remove the estimatedlinear signal path from the gain imbalance gPD(f) and the    phase skewφPD(f) to obtain post-modulation gain imbalance gPM(f) and post-   modulation phase skew φPM(f) by using the transform described in“Altering    the Gain Imbalance and Phase Skew through a Linear System”,with    H(f)=exp(−jθ(0))/m(f) and H(−f)=exp(−jθ(0))/m(−f).    Note: If−f has not yet been visited by the frequency offset loop, use the m(−f)   computed in the previous quality iteration k−1.   10.  Generateupdate for transmitter gain imbalance gT and transmitter phase   skew φTaccording to:         gT(f, k+1)← gT(f, k)*gPM(f) and         φT(f,k+1)←φT(f, k)+φPM(f).   11.  Calculate the image rejection IR(f) fromthe post-modulation gain imbalance   gPM(f) and post-modulation phaseskew φPM(f) based on equation (4.15)  Endfor  k ← k+1  Calculate qualitymeasure Q = maximum of −IR(f) over all values of f.  (A more  negativevalue of IR(f) corresponds to higher quality. Thus, the negative ofIR(f)  corresponds to quality at frequency f. Q is the maximum ofquality over the frequency  band.) End While

APPENDIX B Iterative Estimation of Transmitter Impairments Using OffsetLOs - Optimized 1. Configure the receiver and transmitter so that thedifference between the receiver's local oscillator frequency LO_(RX) andthe transmitter's local oscillator frequency LO_(TX) is equal to aselected value ΔLO:              LO_(Rx)−LO_(TX)=ΔLO. The selected valueis a non-zero fraction (e.g., a small fraction) of the instantaneousbandwidth of the transmitter. The two local oscillators are phaselocked. 2. Connect the transmitter's output to the receiver's input. 3.Estimate the DC scaling m(0) and DC rotation θ(0) of the signal pathbetween transmitter's I/Q modulator and the receiver's I/Q demodulatorusing the algorithm in the section “Calculating the Mapping Between andRX and TX”. This estimation involves the following steps.  3A. Apply azero stimulus signal as input to the transmitter's I/Q modulator.  3B.Capture the response signal z_(A)(n) at the output of the receiver's I/Qdemodulator.  3C. Frequency shift the response signal z_(A)(n) by amountΔLO to obtain a frequency-  shifted signal FSz_(A)(n).  3D. Apply a DCtest vector as input to the I/Q modulator.  3E. Capture the responsesignal z_(C)(n) at the output of the I/Q demodulator.  3F. Frequencyshift the response signal z_(B)(n) by amount ΔLO to obtain a frequency- shifted signal FSz_(B)(n).  3G. Compute the DC scaling m(0) and the DCrotation θ(0) based on the frequency  shifted signal FSz_(A)(n), thefrequency shifted signal FSz_(B)(n) and the DC test vector as  describedin the section “Calculating the Mapping between the RX and TX”. For thebest results, apply a tone K to the transmitter's I/Q modulator inaddition to the DC test vector. The tone K is applied because theleakage can be sensitive to in-band power. The tone K is applied at somefrequency in the instantaneous bandwidth other than DC. Note: Thefrequency-shifting operations may be performed using a signal FS(n)whose phase is continuous in time and runs at rate ΔLO. For example,FS(n) may have the form:        FS(n) = exp{j2π(ΔLO/ADC_SampleRate)n}.The frequency shift operation may be implemented according to therelation:        FSz(n) = z(n)FS(n), where z(n) is a signal to befrequency shifted. In one embodiment, the frequency shift operation maybe implemented in an FPGA of the receiver. The frequency shiftingoperation may execute at the sample rate of the receiver's ADC, i.e.,may generate a new output value FSz(n) for each new ADC data vectorz(n). Thus, the ADC sample clock may be provided as an input to theFPGA. The phase-continuity of the signal FS would then be guaranteed bythe phase continuity of the ADC sample clock. The ADC sample clock isphase locked to the local oscillators. In an alternative embodiment, thefrequency shift operation may be performed in software. The presentiterative method involves repeated acquisitions of the signal z(n) fromthe I/Q demodulator. Thus, to implement the phase-continuity of thesignal FS, software is supplied with information regarding the timedifference between the start of the present acquisition and the start ofthe first acquisition (or the start of the previous acquisition). Forexample, software may be supplied with the time of the first sample z(0)of the present acquisition relative to the time of the first sample z(0)of the first acquisition. Let m be defined as a continuously-runningsample count and n be the sample count of the present acquisition. Thus, m=0 corresponds to n=0 for the first acquisition of z(n). Thenthe phase-continuous frequency shift signal FS(m) can be represented as:      FS(m)=exp{j2π(ΔLO/ADC_SampleRate)m}. Let k be defined by thesample distance between the current acquisition and the firstacquisition for the first sample z(0). Then     FS(m)=FS(k+n)=exp{j2π(ΔLO/ADC_SampleRate)(k+n)}. Now FSz(n) cancomputed from the expression       FSz(n)=FS(k+n)z(n)=FS(n)z(n)FSOffset, where       FS(n)=exp{j2π(ΔLO/ADC_SampleRate)n}     FSOffset=FS(k)=exp{j2π(ΔLO/ADC_SampleRate)k}. Note that k will onlychange from one acquisition to the next. For each positive tonefrequency offset f = Δf to NΔf step Δf, subject to restrictionsdescribed in the “Restrictions” section.  k ← 0  For S element of {1,−1}  Do while ( −Image_Rejection for tone frequency offset v=S*f is smallerthan   threshold):    4.  Compute α and β coefficients forpre-correction circuitry based at least on    the best availableestimate for transmitter impairments at frequency v, as    follows:    If f = Δf      If k=0      if S=1:       Set gT(v,0)←1 andφT(v,0)←0.       Set pre-correction coefficients α and β to implementthe identity       map (i.e., straight pass-through): α←0 and β←1.     if S=−1:       gT(v,0)← gT(−v,∞).       φT(v,0)← φT(−v,∞).       Ingeneral, the notation gT(x,∞) and φT(x,∞) represent respectively      the converged estimates of gT and φT resulting from the final k      iteration at previously-visited frequency x.       Compute α and βfor a traditional single-point compensation based       on gT(v,0) andφT(v,0).     Else k>0      if S=1: Compute α and β for a traditionalsingle-point compensation      based on gT(v,k) and φT(v,k).      ifS=−1 : Compute α and β for true single-point correction based on     gT(v,k) and φT(v, k), gT(−v,∞) and φT(−v,∞).     End If    Else(f > Δf)     If k=0      gT(v,0)← gT(v−S*Δf, ∞)      φT(v,0)← φT(v−S*Δf,∞)      Compute α and β for true single-point correction based on thebest      available estimate of the transmitter impairments at v and −v,e.g., as      follows.      if S=1: compute α and β for truesingle-point correction based on      gT(v−Δf, ∞), φT(v−Δf, ∞),gT(−v+Δf, ∞), φT(−v+Δf, ∞)      if S=−1: compute α and β for truesingle-point correction based on      gT(v+Δf, ∞), φT(v+Δf, ∞), gT(−v,∞), φT(−v, ∞)     Else k>0      if S=1, compute α and β for truesingle-point correction based on      gT(v, k), φT(v, k), gT(−v+Δf, ∞),φT(−v+Δf, ∞)      if S=−1, compute α and β for true single-pointcorrection based on      gT(v, k), φT(v, k), gT(−v,∞), φT(−v, ∞)    EndIf   5. Configure the pre-correction circuitry to use the computedvalues α and β.   6. Apply a complex exponential signal u(n)=exp(j2πvn)to the inputs of the pre-   correction circuitry.   7A.  Measure complexbaseband signal z(n) at the output of the receiver's I/Q   demodulator.  7B.  This step is optional.    Remove the receiver's I/Q impairmentsfrom the complex baseband signal    z(n) to obtain a modified complexsignal. For example, this removal may    involve filtering the complexbaseband signal with a 2×2 matrix of digital    filters, or, multiplyingthe complex baseband signal by a 2×2 constant matrix,    as describedabove in the section “Determination of Transmitter I/Q    Impairmentswith Offset LOs”.   7C.  Apply a frequency shift equal to ΔLO that isphase continuous (as   described above) to the signal z(n) in order toobtain frequency-shifted signal   FSz(n). If step 7B has been performed,the frequency shift is applied to the   modified complex signal.   8.Determine raw gain imbalance gFSz(v) and raw phase skew φFSz(v) at  frequency v based on the complex baseband signal FSz(n) using thecalculation   method described in the “Precise Measurement Technique”section.   9. Remove the best current estimate of the signal path(between the transmitter's   I/Q modulator and the receiver's I/Qdemodulator) from the raw gain imbalance   gFSz(v) and the raw phaseskew φFSz(v) to obtain estimated post-modulation   gain imbalance gPM(v)and post-modulation phase skew φPM(v). m(0) and   θ(0) will provide abasic estimate of the signal path. Better estimates will   increase therate of convergence. For example, step 9 may be implemented as  follows.    If f=Δf     Remove the estimated DC scaling m(0) and DCrotation θ(0) from the raw     gain imbalance gFSz(v) and the raw phaseskew φFSz(v) to obtain the     estimated post-modulation gain imbalancegPM(v) and post-modulation     phase skew φPM(v) using the transformdescribed in “Altering the Gain     Imbalance and Phase Skew through aLinear System”, with     H(v) = exp(−jθ(0))/m(0) and H(−v) =exp(−jθ(0))/m(0).    Else f>Δf     Compute scaling m(v) at tonefrequency v based on the signal FSz(n) of     step 7C. The scaling m(v)may be determined by computing the     magnitude of the frequencycomponent at frequency v in the complex     signal FSz(n), as explainedin the “Precise Measurement Technique”     section, especially inequation 6.6.     (Note: In an alternative embodiment, the measurementof z(n) is     synchronized with the generation of the tone u(n), e.g.,by using a trigger     signal that is shared between the transmitter andreceiver, e.g., a trigger     generated by a controller device. In thiscase, rotation θ(v) may be     measured in addition to scaling m(v).)    Remove the estimated linear signal path from the raw gain imbalance    gFSz(v) and the raw phase skew φFSz(v) to obtain the estimated post-    modulation gain imbalance gPM(v) and post-modulation phase skew    φPM(v) by using the transform described in “Altering the GainImbalance     and Phase Skew through a Linear System” with H(v) =exp(−jθ(0))/m(v)     and H(−v)=exp(−jθ(0))/m_(BAE)(−v), wherem_(BAE)(−v) is the best available     estimate for scaling m(−v).     If S=1: m_(BAE)(−v)=m(−v+Δf, ∞)      If S=−1 : m_(BAE)(−v)=m(−v,∞).     In general, the notation m(x,∞) denotes the scaling m(x) computedin      the final k iteration of previously-visited frequency x.    10. Generate update for transmitter gain imbalance gT and transmitter phase   skew φT according to:         gT(v, k+1)← gT(v, k)*gPM(v) and        φT(v, k+1)←φT(v, k)+φPM(v).    11.  Calculate the imagerejection IR(v) from the post-modulation gain    imbalance gPM(v) andpost-modulation phase skew φPM(v) based on equation    4.15    k ← k+1  EndDo  EndFor S element of {1,−1} Endfor

APPENDIX C Iterative Estimation of Transmitter Impairment Using SharedLO - Optimized 1. Measure the gain imbalance gR and phase skew φR of thereceiver for each in-band offset frequency fat which the transmitter'sgain imbalance gT and phase skew φT are to be measured. For each f,direct a tone generator to generate a tone at frequency v=f_(LO)+f,where f_(LO) is the LO frequency, apply the tone to the receiver'sinput, and capture the complex baseband sequence z(n) at the output ofthe receiver's I/Q demodulator. The gain imbalance gR and phase skew φRare computed as described in the “Precise Measurement Technique”section. 2. Configure the receiver and transmitter so that they use thesame LO frequency f_(LO). If the receiver and transmitter use twodifferent LO circuits, tune the transmitter so that its LO is phaselocked to the same reference. Therefore, the frequency of thetransmitter and the frequency of the receiver are both f_(LO). 3.Connect the transmitter's output to the receiver's input. 4. Estimatethe DC scaling m(0) and DC rotation θ(0) of the signal path betweentransmitter's I/Q modulator and the receiver's I/Q demodulator by usingthe algorithm in the section “Calculating the Mapping Between and RX andTX”. For the best results, apply a tone K to the transmitter's I/Qmodulator in addition to the DC test vector. For each positive frequencyoffset f = Δf to NΔf step Δf.  For S element of {1,−1}   k ← 0   Dowhile ( −Image_Rejection for frequency offset v=S*f is smaller thanthreshold):    5A. Compute α and β coefficients for pre-correctioncircuitry based at least on    the best available estimate fortransmitter impairments at frequency v.     If f = Δf      If k=0      if S=1:        Set gT(v,0)←1 and φT(v,0)←0, and set pre-correctioncoefficients        α and β to implement the identity map (i.e.,straight pass-through):        α←0 and β←1.       if S=−1:        SetgT(v,0)← gT(−v,∞), φT(v,0)← φT(−v,∞), and compute α and β        for atraditional single-point compensation based on gT(v,0) and       φT(v,0).      Else k>0       if S=1: Compute α and β for atraditional single-point compensation       based on gT(v,k) and φT(v,k)      if S=−1: Compute α and β for true single-point compensation basedon       gT(v,k) and φT(v, k), gT(−v,∞) and φT(−v,∞)      End If    Else (f > Δf)      If k=0       gT(v,0)← gT(v−S*Δf, ∞) and φT(v,0)←φT(v−S*Δf, ∞)       Compute α and β for true single-point correctionbased on the best       available estimate of the transmitterimpairments at v and −v, e.g., as       follows.       if S=1:       compute α and β for true single-point correction based on       gT(v−Δf, ∞), φT(v−Δf, ∞), gT(−v+Δf, ∞), φT(−v+Δf, ∞)       ifS=−1:        compute α and β for true single-point correction based on       gT(v+Δf, ∞), φT(v+Δf, ∞), gT(−v, ∞), φT(−v, ∞)      Else k>0      if S=1:        compute α and β for true single-point correctionbased on gT(v, k),        φT(v, k), gT(−v+Δf, ∞), φT(−v+Δf, ∞)       ifS=−1:        compute α and β for true single-point correction based ongT(v, k),        φT(v, k), gT(−v,∞), φT(−v, ∞)      End If     End If   5B. Configure the pre-correction circuitry to use the computed valuesα and β.    6. Apply a complex exponential signal u(n)=exp(j2πvn) to theinputs of the pre-    correction circuitry.    7A.  Measure complexbaseband signal z(n) at the output of the receiver's I/Q    demodulator.   7B.  Determine raw gain imbalance gz(v) and raw phase skew φz(v)based on    the complex baseband signal z(n) using the calculationmethod in the section    “Precise Measurement Technique”.    8. Removethe receiver's gain imbalance gR(v) and phase skew φR(v) from the    rawgain imbalance gz(v) and raw phase skew φz(v) to obtain pre-demodulation   gain imbalance gPD(v) and pre-demodulation phase skew φPD(v). Thereare a    number of ways to accomplish this removal, including amathematical    transformation method and a filtering method, asdescribed above in connection    with method 4400. The mathematicaltransformation method is described in the    section “Removing ReceiverImpairments from Measured Output Impairments”.    9. Remove the bestcurrent estimate of the signal path between the transmitter's    I/Qmodulator and the receiver's I/Q demodulator. m(0) and θ(0) will providea    basic estimate. Better estimates will increase the rate ofconvergence. For    example, step 9 may be implemented as follows.    Iff = Δf:     Remove the estimated DC scaling m(0) and DC rotation θ(0)from the gain     imbalance gPD(v) and the phase skew φPD(v) to obtainthe estimated post-     modulation gain imbalance gPM(v) andpost-modulation phase skew φPM(v)     using the transform described in“Altering the Gain Imbalance and Phase     Skew through a Linear System”with H(v) and H(−v) set equal to      exp(−jθ(0))/ m(0).    Else f>Δf    Compute scaling m(v) at tone frequency v based on the complexbaseband     signal z(n) of step 7A. The scaling m(v) may be determinedby computing     the magnitude of the frequency component at frequency vin the complex     signal z(n), as explained in the “Precise MeasurementTechnique” section,     especially in equation 6.6.     (Note: In analternative embodiment, the measurement of z(n) is     synchronized withthe generation of the tone u(n), e.g., by using a trigger     signalthat is shared between the transmitter and receiver, e.g., a trigger    generated by a controller device. In this case, rotation θ(v) may bemeasured     in addition to scaling m(v).)     Remove the estimatedlinear signal path from the gain imbalance gPD(v) and     the phase skewφPD(v) to obtain the estimated post-modulation gain     imbalance gPM(v)and post-modulation phase skew φPM(v) by using the     transformdescribed in “Altering the Gain Imbalance and Phase Skew     through aLinear System” with H(v)=exp(−jθ(0))/m(v) and    H(−v)=exp(−jθ(0))/m_(BAE)(−v), where m_(BAE)(−v) is the bestavailable estimate for     scaling m(−v).      If S=1:m_(BAE)(−v)=m(−v+Δf, ∞)      If S=−1: m_(BAE)(−v)=m(−v,∞).      Ingeneral, the notation m(x,∞) denotes the scaling m(x) computed in the     final k iteration at previously-visited frequency x.     10. Generate update for transmitter gain imbalance gT and transmitter    phase skew φT according to:           gT(v, k+1)← gT(v, k)*gPM(v)and           φf(v, k+1)←φT(v, k)+φPM(v).    11.  Calculate the imagerejection IR(v) from the post-modulation gain    imbalance gPM(v) andpost-modulation phase skew φPM(v) based on equation    4.15    k ← k+1  EndDo  EndFor S element of {1,−1} Endfor

What is claimed is:
 1. A computer-implemented method for computing I/Q impairments at a complex output of an electrical system based on I/Q impairments at a complex input of the electrical system, the method comprising: computing a first spectrum based on: a model spectrum H(f) that models the electrical system, an input gain imbalance spectrum g(f) that characterizes gain imbalance at the complex input, and an input phase skew spectrum φ(f) that characterizes phase skew at the complex input; computing a second spectrum based on: a frequency-reflected version H(−f) of the model spectrum H(f), the input gain imbalance spectrum g(f), and the input phase skew spectrum φ(f); computing a sum of the first spectrum and the second spectrum, and a difference of the first spectrum and the second spectrum; computing an output gain imbalance spectrum and an output phase skew spectrum based on real and imaginary parts of the sum, and real and imaginary parts of the difference, wherein the output gain imbalance spectrum and the output phase skew spectrum respectively characterize gain imbalance and phase skew at the complex output; and storing the output gain imbalance spectrum and the output phase skew spectrum in a memory.
 2. The method of claim 1, wherein the model spectrum H(f) represents an inverse of a signal path, wherein the signal path is a path from an I/Q modulator of a transmitter to a demodulator of a receiver, wherein the input gain imbalance spectrum and the input phase skew spectrum represent gain imbalance and phase skew at an output of the demodulator, wherein the output gain imbalance spectrum and the output phase skew spectrum represent gain imbalance and phase skew at an output of the I/Q modulator.
 3. The method of claim 2, further comprising: computing an inverse of a spectrum of the signal path to determine the model spectrum H(f).
 4. The method of claim 2, wherein the model spectrum H(f) is based on a DC scaling and a DC rotation of the signal path.
 5. The method of claim 4, wherein the DC scaling is determined by: capturing a first response signal from the demodulator in response to a zero signal being supplied as input to the I/Q modulator; capturing a second response signal from the demodulator in response to a constant signal being supplied as input to the I/Q modulator, wherein the constant signal is equal to a non-zero complex constant; averaging the first response signal to obtain a first average and averaging the second response signal to obtain a second average; computing a difference between the second average and the first average; and computing the DC scaling based on the difference and the non-zero complex constant.
 6. The method of claim 1, wherein the model spectrum H(f) represents an inverse of a signal path, wherein the signal path is a path from an I/Q modulator of a transmitter to a demodulator of a receiver, wherein the input gain imbalance spectrum and the input phase skew spectrum represent gain imbalance and phase skew at an input of the demodulator, wherein the output gain imbalance spectrum and the output phase skew spectrum represent gain imbalance and phase skew at an output of the I/Q modulator.
 7. The method of claim 1, wherein the method is performed in a programmable hardware element.
 8. The method of claim 1, wherein the method is performed by a processor in response to execution of program instructions.
 9. The method of claim 1, further comprising: measuring the input gain imbalance spectrum g(f) and the input phase skew spectrum φ(f) of an electronic device.
 10. A non-transitory computer-accessible memory medium for computing I/Q impairments at a complex output of an electrical system based on I/Q impairments at a complex input of the electrical system, wherein the memory medium stores program instructions, wherein the program instructions, when executed by a computer system, cause the computer system to: compute a first spectrum based on: a model spectrum H(f) that models the electrical system, an input gain imbalance spectrum g(f) that characterizes gain imbalance at the complex input, and an input phase skew spectrum φ(f) that characterizes phase skew at the complex input; compute a second spectrum based on: a frequency-reflected version H(−f) of the model spectrum H(f), the input gain imbalance spectrum g(f), and the input phase skew spectrum φ(f); compute a sum of the first spectrum and the second spectrum, and a difference of the first spectrum and the second spectrum; compute an output gain imbalance spectrum and an output phase skew spectrum based on real and imaginary parts of the sum, and real and imaginary parts of the difference, wherein the output gain imbalance spectrum and the output phase skew spectrum respectively characterize gain imbalance and phase skew at the complex output; and store the output gain imbalance spectrum and the output phase skew spectrum in a memory.
 11. The non-transitory computer-accessible memory medium of claim 10, wherein the model spectrum H(f) represents an inverse of a signal path, wherein the signal path is a path from an I/Q modulator of a transmitter to a demodulator of a receiver, wherein the input gain imbalance spectrum and the input phase skew spectrum represent gain imbalance and phase skew at an output of the demodulator, wherein the output gain imbalance spectrum and the output phase skew spectrum represent gain imbalance and phase skew at an output of the I/Q modulator.
 12. The non-transitory computer-accessible memory medium of claim 11, wherein the program instructions, when executed by the computer system, further cause the computer system to: compute an inverse of a spectrum of the signal path to determine the model spectrum H(f).
 13. The non-transitory computer-accessible memory medium of claim 11, wherein the model spectrum H(f) is based on a DC scaling and a DC rotation of the signal path.
 14. The non-transitory computer-accessible memory medium of claim 11, wherein the model spectrum H(f) represents an inverse of a signal path, wherein the signal path is a path from an I/Q modulator of a transmitter to a demodulator of a receiver, wherein the input gain imbalance spectrum and the input phase skew spectrum represent gain imbalance and phase skew at an input of the demodulator, wherein the output gain imbalance spectrum and the output phase skew spectrum represent gain imbalance and phase skew at an output of the I/Q modulator.
 15. The non-transitory computer-accessible memory medium of claim 10, wherein the program instructions, when executed by the computer system, further cause the computer system to: measure the input gain imbalance spectrum g(f) and the input phase skew spectrum φ(f) of an electronic device.
 16. A computer system for computing I/Q impairments at a complex output of an electrical system based on I/Q impairments at a complex input of the electrical system, the computer system comprising: a processor; and memory storing program instructions, wherein the program instructions, when executed by the processor, cause the processor to: compute a first spectrum based on: a model spectrum H(f) that models the electrical system, an input gain imbalance spectrum g(f) that characterizes gain imbalance at the complex input, and an input phase skew spectrum φ(f) that characterizes phase skew at the complex input; compute a second spectrum based on: a frequency-reflected version H(−f) of the model spectrum H(f), the input gain imbalance spectrum g(f), and the input phase skew spectrum φ(f); compute a sum of the first spectrum and the second spectrum, and a difference of the first spectrum and the second spectrum; compute an output gain imbalance spectrum and an output phase skew spectrum based on real and imaginary parts of the sum, and real and imaginary parts of the difference, wherein the output gain imbalance spectrum and the output phase skew spectrum respectively characterize gain imbalance and phase skew at the complex output; and store the output gain imbalance spectrum and the output phase skew spectrum in a memory.
 17. The computer system of claim 16, wherein the model spectrum H(f) represents an inverse of a signal path, wherein the signal path is a path from an I/Q modulator of a transmitter to a demodulator of a receiver, wherein the input gain imbalance spectrum and the input phase skew spectrum represent gain imbalance and phase skew at an output of the demodulator, wherein the output gain imbalance spectrum and the output phase skew spectrum represent gain imbalance and phase skew at an output of the I/Q modulator.
 18. The computer system of claim 17, wherein the program instructions, when executed by the processor, further cause the processor to: compute an inverse of a spectrum of the signal path to determine the model spectrum H(f).
 19. The computer system of claim 17, wherein the model spectrum H(f) is based on a DC scaling and a DC rotation of the signal path.
 20. The computer system of claim 16, wherein the model spectrum H(f) represents an inverse of a signal path, wherein the signal path is a path from an I/Q modulator of a transmitter to a demodulator of a receiver, wherein the input gain imbalance spectrum and the input phase skew spectrum represent a gain imbalance and a phase skew at an input of the demodulator, wherein the output gain imbalance spectrum and the output phase skew spectrum represent a gain imbalance and a phase skew at an output of the I/Q modulator. 